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PEIMARY   ARITHMETIC 


NUMBER  STUDIES  '-'  '  ' 


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F0i2  THE  SECOND,  THIRD,  AND  FOURTH   GRADES 


BY 

A.    R.    HORNBROOK,   A.M. 

TEACHER   IN   THE   PUBLIC    SCHOOLS   OF   EVANSVILLE,    IND. 


-oo'ioioo- 


NEW  YORK  •:•  CINCINNATI  :•  CHICAGO 

AMERICAN    BOOK    COMPANY 


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Copyright,  1898,  by 
A.   E.   HORNBROOK. 


PBIM.  ABITH.  —  HORNBROOK. 


VafiFV. 


PREFACE 


The  progress  of  a  beginner  in  arithmetic  is  of  a 
desirable  kind  when  it  involves  a  snccession  of  insights 
into  the  relations  of  numbers  and  an  increase  of  expert- 
ness  in  dealing  with  them.  It  is  the  aim  of  this  first 
book  to  secure  these  ends.  Its  material  has  been  chosen 
with  careful  reference  to  the  development  of  the  number 
sense  of  little  children  as  observed  by  the  author  and  as 
reported  by  many  other  observers. 

It  is  believed  that  when  a  child  realizes  the  meanings 
of  the  first  ten  number  names,  has  learned  to  make  com- 
binations within  10,  and  is  able  to  count  to  100,  he  is 
ready  to  take  up  the  first  hundred  as  an  aggregation  of 
tens,  to  consider  other  numbers  as  aggregations,  and  to 
discover  their  relations.     At  that  point  this  book  begins. 

The  use  of  diagrams  called  "  number  tables  "  as  a  con- 
crete basis  for  the  child's  thinking  while  he  is  getting  his 
first  ideas  of  the  facts  of  the  addition  and  multiplication 
tables  is  a  distinctive  feature  of  the  work.  Children 
readily  learn  from  a  number  table  like  that  on  page  14 
such  facts  as  "5  tens  =  50."  The  five  columns  of  num- 
bers are  as  concrete  to  them  as  five  sticks,  and  the  figures 
"•50  "  at  the  end  of  the  fifth  column  make  them  much 
more  suggestive.  Much  of  the  work  given  in  this  book 
would  be  entirely  too  difficult  for  the  children  for  whom 
it  is  intended  if  it  lacked  the  basis  of  the  measurements 
of  the  number  tables.     -    .  .    -   i  /« 

54  *^'*}44 


4  PREFACE 

The  treatment  of  numbers  used  in  this  book  leads  to 
the  presentation  of  the  multiplication  tables  in  an  order 
different  from  that  usually  followed,  and  more  economical 
of  children's  time  and  effort.  10,  "the  master  key  of 
number,"  under  the  decimal  system,  is  presented  first 
with  its  multiples.  The  child's  instinct  for  grouping 
by  pairs  is  next  utilized  by  giving  the  table  of  twos. 
Work  in  addition  and  subtraction  follows  in  which  the 
relations  of  numbers  to  10  and  to  2  are  frequently  brought 
to  mind.  By  objective  work  in  feet  and  yards  illustrat- 
ing_  combinations  in  addition,  the  pupil  gains  a  knowledge 
of  multiples  of  3.  The  smaller  multiples  of  4  are  learned 
by  similar  work  upon  quarts  and  gallons,  pecks  and 
bushels.  The  fives  as  a  subdivision  of  the  tens  are  pre- 
sented in  the  next  chapter,  and  in  order  that  the  child 
may  have  time  to  become  familiar  with  the  multiples  of 
5,  most  of  the  work  of  that  chapter  relates  to  them. 
The  child  has  been  dealing  with  10  and  its  divisions,  and 
has  had  much  practice  in  combining  10  with  other  num- 
bers. To  learn  the  table  of  elevens  is  an  easy  task  for 
him.  One  little  fellow  remarked,  "Learning  the  table 
of  elevens  is  just  like  going  down  stairs,  and  you  can 
always  tell  what  step  you  are  on.  The  first  step  is  made 
of  I's  and  the  second  step  is  made  of  2's,  and  it  is  that 
way  all  the  way  down."  A  glance  at  the  oblique  line 
made  by  the  multiples  of  11  in  the  number  table  on  page 
114  will  explain  his  remark.  9,  as  a  departure  from  10 
on  the  other  side,  is  next  given.  The  table  of  nines  is 
reenforced  by  that  of  the  threes,  which  receives  formal 
treatment  in  the  next  chapter.  The  treatment  of  8  is 
followed  by  that  of  its  subdivision  4. 

Work  in  fractions,  which  is  generally  so  successful  in 
first  grades,  is  continued  throughout  the  book  in  connec- 


PREFACE  5 

tion  with  simple  geometric  forms,  and  leads  naturally  to 
the  recognition  of  ratios. 

Only  the  rare,  precocious  child  is  able  to  found  a 
process  upon  a  course  of  reasoning,  however  clearly  it 
may  be  presented.  For  that  reason,  only  those  processes 
of  written  work  that  can  be  based  upon  the  child's  intui- 
tions of  number  are  accounted  for  ;  others  are  given 
simply  as  processes  leading  to  desired  results,  without 
any  attempt  at  forcing  a  knowledge  of  the  underlying 
principles  into  the  immature  mind.  The  child  is  led  to 
construct,  to  observe,  to  report,  and  to  remember,  but  the 
reasoning  required  of  him  in  the  first  book  is  limited  to 
simple  inferences. 

Formal  analysis,  that  most  effective  deadener  of  the 
mathematical  sense  of  little  children,  has  been  omitted. 
The  successful  teacher  knows  how  to  stimulate  the  ex- 
pression of  the  child's  own  insights  into  iiumber  by  light, 
skillful  touches  upon  his  mind  in  easy  conversational 
exercises. 

The  development  of  the  plan  of  the  work  is  indicated 
by  many  notes  to  the  teacher. 

To  the  many  primary  teachers  who  have  kindly  con- 
tributed the  results  of  their  schoolroom  experiences,  the 
author  offers  grateful  acknowledgments. 


CONTENTS 


CHAPTER 

I.     Squares  —  Counting 

II.     Tens 

Cents  and  Dimes 
Written  Addition 
Written  Subtraction 
Tens  and  Units 
Roman  Numeral  X 

III.  Twos      . 

Even  Numbers  . 

Foot  and  Inch  . 

Halves 

Quart  and  Pint 

Horizontal  Line 

Triangle    . 

Thirds 

Fourths 

Vertical  Line     . 

Roman  Numeral  I 

IV.  Addition 

Sum  . 

Yard 

Rectangle 

Thousands 

Gallon 

Perimeter 

Decimal  Point  . 

Roman  Numerals  V,  L,  and  C 

Peck 

V.     Subtraction 
Difference 
Minuend   . 
Subtrahend 
Pound  and  Ounce 

VI.     Applications  of  Addition  and 
Industrial  Problems 
Days  in  Months 
Odd  Numbers    . 

VII.     Fives      . 

Equilateral  Triangles 
Roman  Numerals  D  and  M 
Quotient   .... 


Subtraction 


PAGE 

9 

14 

17 
21 
26 
32 
34 

35 
36 
39 
40 
41 
42 
■  44 
46 
48 
48 
49 

51 
53 
56 
64 
64 
66 
71 
74 
81,  82 
79 

83 
84 
89 
92 
93 

95 

96 

98 

100 

104 

108 
108,  113 
.     Ill 


75, 


The  table  of  contents  shows  the  chapter  in  which  a  subject  first  appears. 
Each  subject  reappears  in  succeeding  chapters. 

7 


8  CONTENTS 

CHAPTER  PAGE 

VIII.     Elevens 114 

Written  Multiplication 120 

Product 120 

IX.     Nines 122 

Multiplier 123 

Square  Yard 124 

Square  of  a  Number 125 

Divisor 130 

X.  Threes 132 

Multiplicand 133 

Parallel  Lines 135 

Trapezoid 136 

Khombus  . 137 

Ratio 139 

XI.  Eights 142 

Denominator 147 

Quart  and  Peck 148 

Short  Division 148 

Dividend .  149 

Perpendiculars 149 

Area  of  Eight  Triangle 151 

XII.     Fours 154 

Numerator 159 

Square  Prism 164 

Partial  Products 165 

Ton 166 

XIII.  Sevens 167 

Factors 169 

Compound  Fractions 173 

XIV.  Sixes 180 

Rod 186 

Hexagon 188 

Interest 190 

XV.     Twelves 192 

Square  Foot 195 

Long  Division 199 

Cubic  Foot 202 

Common  Multiple      . 203 

XVI.     Review 205 

Average 206 

Common  Divisor 211 

Addition  of  Compound  Denominate  Numbers  .         .         .221 

Subtraction  of  Compound  Denominate  Numbers      .         .  222 

Multiplication  of  Compound  Denominate  Numbers          .  223 

Per  cent 227 

Bills 230 

XVIL     Fractions 232 


1   5  1  1       i  > 


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1      ■> 


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5      1  1 

1      J     J 


ELEMENTARY  ARITHMETIC 


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CHAPTER   I 


SQUARES  —  COUNTING 

Inch  squares  cut  fi'om  white  paper  should  be  prepared  in  such 
abundance  that  each  child  may  have  enough  to  make  the  figures  given 
in  this  chapter.  Draw  the  figures  on  the  blackboard  and  give  the 
work  orally  at  first. 

1.  How  many  squares  in  Fig.  1  ? 

2.  Place  squares  in  a  column  like  Fig.  1. 

3.  If  we  call  the  square  at  the  top  the  first 
square,  and  the  next  one  the  second,  and  so  on, 
how  shall  we  number  the  last  square?  How  shall 
we  number  the  next  to  the  last  square? 

4.  Show  the  fourth  square  in  your  column. 
Show  the  sixth  square,  the  ninth  square,  the  third 
square,  the  seventh  square,  the  fifth  square,  the 
eighth  square. 

5.  Push  the  lowest  two  squares  away.  How 
many  squares  are  left? 

6.  Make  the  column  whole  again.  Take  away 
four  squares  at  the  lower  end  of  the  column. 
How  many  squares  are  left? 

7.  Divide  the  column  into  two  equal  parts. 
How  many  squares  in  each  part? 

8.  Take  away  from  the  whole  column  six 
squares,  and  tell  how  many  are  left. 

9 


Fig.  1 


<         ( 

i 


10 


SQUARES  —  COUNTING 


9.    Take  away  from  tlie  whole  column  three  squares, 

and  tell  how  many  are  left. 

Give  exercises  in  parting  and  wholing  the  column  of  squares  until 
the  combinations  up  to  10  are  thoroughly  reviewed. 

10.  Place  squares  as  in  Fig.  2.  How  is  it 
different  from  Fig.  1?  How  many  squares 
in  Fig.  2?     10  and  2  are  how  many? 

11.  Add  2  more  squares  to  the  short  col- 
umn. Tell  how  many  squares  there  are  now 
in  tlie  short  column.  How  many  in  the 
whole  figure?     10  and  4  are  how  many? 

12.  Add  2  more  squares.  How  many 
squares  in  the  short  column  now?  How 
many  in  tlie  whole  figure?  10  and  6  are 
how  many? 

13.  Add  2  more  squares.  How  many 
squares  in  the  short  column?  How  many 
in  tlie  whole  figure?  10  and  8  are  how 
many? 

14.  Add  squares  to  the  short  column  until 
the  columns  are  equal.  How  many  did  you 
add?  How  many  squares  in  each  column? 
How  many  in  the  whole  figure?     10  and  10 

Pig.  2  are  how  many  ? 

15.  Take  away  the  last  two  squares  from  the  figure  you 
have.  How  many  are  left?  2  from  20  leave  liow  many  ? 
Take  away  2  more  and  tell  how  many  are  left.  2  from  18 
leave  how  many  ? 

16.  Keep  taking  away  two  more  and  telling  how  many 
are  left  until  the  right-hand  column  is  all  gone.  2  from 
16  leave  how  many?  2  from  14  leave  how  many?  2  from 
12  leave  how  many? 


SQUAFiES  —  COUNTING 


11 


Give  similar  exercises  on  successive  days  until  these  facts  of  number 
measurement  have  been  called  into  the  consciousness  of  the  children 
so  often  and  so  clearly  that  they  have  become  a  part  of  their  mental 
property.  Do  not  let  them  memorize  number  statements  such  as 
"  10  and  2  are  12  "  until  it  is  evident  that  their  statements  are  sup- 
ported by  their  own  perceptions  of  number  truths. 


17.  Put  the   20   squares   back   into   2   equal   columns. 
Number  them  as  in  Fig.  3. 

18.  Find    the    17th    square.     How   many 
squares  in  this  figure  come  after  the   17th? 
How  many  squares  before  the  17th  square  are 
numbered  in  this  figure  ? 

19.  Find  the  15th  square  and  show  how 
many  squares  come  after  it.  How  many 
squares  come  before  the  15th  square  in  this 
figure  ?  Show  how  many  squares,  in  this 
figure,  come  after  the  13th.  After  the  16th. 
After  the  14th.  After  the  11th.  After  the 
18th.     After  the  12th. 


20. 


3  from  20  =  how  many  ? 

5  from  20  =  how  many  ? 

8  from  20  =  how  many  ? 

4  from  20  =  how  many  ? 

6  from  20  =  how  many 

9  from  20 
2  from  20 

7  from  20  =  how  many 


=  how  many  ? 
=  how  many  ? 


1 

n 

■> 

12 

■  ■> 

o 

113 

4 

U 

5 

15 

G 

16 

7 

17 

8 

18 

9 

19 

10 

20 

Fig.  3 


Let  the  children  separate  the  figure  before  them  into  unequal 
parts  of  their  own  choosing,  telling  how  many  squares  they  take 
away,  and  how  many  are  left. 


12 


SQUARES  —  COUNTING 


21.  Place  squares  as  in  Fig.  4.  How 
many  squares  in  it  ?  How  many  more 
squares  than  in  Fig.  3  ? 

22.  Show  the  20th  square  in  your 
figure. 

Let  the  squares  be  counted  in  the  order  indi- 
cated in  Fig.  3. 

23.  Show  the  21st  square  and  tell  how 
many  squares  come  after  it. 

24.  21  +  2  =  how  many  ? 
20  +  3  =  how  many  ? 

25.  Add  2  squares  to  the  short  column. 
How  many  are  there  now  in  the  short 
column  ?  How  many  in  the  whole  fig- 
ure ?     23  -f-  2  =  how  many  ? 

26.  Add  2  more  squares  and  tell  how 
many  are  in  the  short  column.  How 
many  are  in  the  whole  figure  ?  25  4-  2 
=  how  many  ? 

27.  Add  2  more  squares  and  tell  how 
many  are  in  the  short  column.  Hoav  many  are  in  the 
whole  figure  ?     27  +  2  =  how  many  ? 

28.  Add  squares  enough  to  make  the  short  column  as 
long  as  the  others.  How  many  did  you  add  ?  How 
many  squares  in  your  whole  figure  ? 

29.  Divide  the  whole  figure  yoc  have  made  into  three 
equal  parts.     How  many  squares  in  each  part  ? 

30.  10  4- 10  +  10  =  how  many  ? 

31.  10  from  30  =  how  many  ?     4  from  30  =  how  many  ? 

6  from  30  =  how  many  ?     3  from  30  =  how  many  ? 
5  from  30  =  how  many  ?     7  from  30  =  how  many  ? 


Fig.  4 


SQUARES  —  COUNTING 


13 


32.  Place  squares  to  make  Fig.  5. 
How  many  squares  does  it  take  ? 
30  +  4  =  how  many  ? 

33.  Add  2  squares  to  the  short 
column  and  tell  how  many  squares 
in  it.  How  many  in  the  Avhole 
figure  now  ?     34  -}-  2  =  how  many  ? 

34.  Add  2  more  squares  to  the 
short  column.  How  many  in  it 
now  ?  How  many  in  the  whole 
figure  ?     36  +  2  =  ? 

35.  Add  squares  enough  to  make 
the  short  column  equal  to  the 
others.  How  many  did  3'ou  add  ? 
How  many  are  there  in  the  whole 
figure  ?     38  +  2  =  ? 

36.  How  man}^  columns  in  the 
whole  figure  ?  Separate  the  figure 
into  4  equal  parts.  How  many 
squares  in  each  part  ? 

37.  Put  the  parts  together  again.  Show  the  31st 
square.  How  many  squares  come  after  it  in  the  tigure  ? 
How  many  are  before  it  ? 

38.  Show  the  33d  square  and  tell  how  many  squares 
follow  it ;  the  35th  ;   the  32d  ;  the  36th  ;   the  38th. 

39.  2  from  40  =  ?     4  from  40  =  ?       6  from  40  =  ? 

7  from  40  =  ?     5  from  40  =  '.^       9  from  40  =  ? 

8  from  40  =  ?     3  from  40  =  ?     10  from  40  =  ? 

Add  squares,  a  few  at  a  time,  to  the  figure  on  the  board  and  let  the 
children  count  and  combine  them.  As  each  column  is  completed, 
number  the  last  square.  Continue  this  work  from  day  to  day  until 
the  figure  of  100  squares  is  completed. 


Fig.  5 


CHAPTER  II 

TENS 

Cents  and  Dimes,  Addition  and  Subtraction,  Tens 
AND  Units,  Roman  Numeral  X 

The  questions  upon  the  table  which  immediately  follow  it  are 
designed  to  lead  the  children  to  analyze  it  as  an  object  of  vision  with- 
out reference  to  its  symbolism.  Similar  questions  should  be  given  a 
few  minutes  every  day  until  the  children  are  familiar  with  the  relative 
positions  of  the  numbers. 


NUMBER 

TABLE* 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

6 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

5Q 

6Q 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10    20    30    40    50   60  70   80  90   100 

*  The  number  table  should  be  written  in  large  figures  upon  the  board, 
or  a  chart  should  be  made  of  it.  The  figures  may  be  drawn  with  char- 
coal upon  manila  paper,  or  paiuted  upon  shade  cloth. 

14 


TENS  15 

1.  How  many  columns  of  numbers  in  this  table? 

2.  How  many  numbers  in  each  column? 

3.  Point  out  and  name  the  first  ten  numbers. 

4.  Point  out  and  name  the  second  ten  numbers.  The 
fourth  ten. 

5.  Show  the  tenth  (or  last)  ten  numbers.  The  ninth 
ten.     The  third  ten. 

6.  What  is  the  first  number  of  the  second  ten?  Of 
the  third  ten?     Of  the  tenth  ten?     Of  the  fifth  ten? 

7.  What  is  the  last  number  of  the  first  ten?  Of  the 
third  ten  ?  Of  the  fourth  ten  ?  Of  the  second  ten  ?  Of 
the  ninth  ten  ? 

8.  What  is  the  last  number  of  the  tenth  ten? 

9.  How  many  numbers  are  there  in  the  whole  table  ? 

10.  Point  out  and  name  the  second  number  of  the  first 
ten. 

11.  What  is  the  second  number  of  the  second  ten  ?  Of 
the  fourth  ten  ?     Of  the  tenth  ten  ? 

12.  Show  the  second  number  in  each  ten,  and  tell  what 
figure  it  ends  with. 

13.  Point  out  and  name  the  third  number  of  each  ten, 
and  tell  what  figure  it  ends  with. 

Let  children  point  out  the  corresponding  numbers  in  each  ten  until 
they  see  their  regular  decimal  succession. 

14.  In  the  table  of  numbers,  which  number  is  written 
just  above  the  number  4  ?  Just  above  the  number  14  ? 
24?     44? 

15.  Which  one  is  written  just  below  the  number  14  ? 
Just  below  24  ?     34  ? 

16.  Find  35  and  show  what  number  is  written  just  at 
the  right  of  it.     At  the  left  of  it. 


16  TENS 

17.  What  number  is  written  just  at  the  right  of  41  ? 
At  the  left  of  it  ? 

18.  Begin  at  the  number  8  and  read  toward  the  right, 
naming  every  number. 

19.  Begin  with  number  3,  and  read  until  you  reach  93. 

20.  Begin  with  number  97,  and  read  to  the  left  until 
you  reach  7. 

21.  In  which  column  do  you  find  the  number  25  ?  48  ? 
67  ?     94  ?     79  ? 

22.  40  is  at  the  end  of  which  column  ?     Where  is  80  ? 

23.  Look  at  the  last  number  of  all  the  columns  and  tell 
what  figure  each  number  ends  with. 

24.  Name  the  next  to  the  last  number  of  each  column. 
What  figure  is  the  same  in  each  ? 

25.  Name  all  the  numbers  in  the  table  that  end  in  7. 
In  5.     In  3.     In  8. 

26.  21  is  at  the  beginning  of  which  column  ?  Where 
is  51  ?     71  ?     91  ? 

27.  Name  all  the  numbers  in  the  table  that  end  in  4. 

28.  Point  out  the  12th  number.  The  22d.  The  32d. 
The  42d.  The  16th.  The  26th.  The  36th.  The  46th. 
The  56th. 

29.  How  many  numbers  in  the  first  two  tens  ? 

30.  10  numbers  and  10  numbers  are  how  many  num- 
bers? 

31.  20  and  10  are  how  many  ? 

32.  30  and  10  are  how  many  ? 

33.  30  and  10  =  how  many  ?      50  and  10  =  how  many  ? 
60  and  10  =  h()^y  many  ?      70  +  10  =  ?     90  +  10  =  ? 


TEN8  17 

The  teacher  should  provide  herself  with  actual  money,  consisting 
of  dimes  and  cents,  with  which  to  illustrate  the  following  work.  Give 
much  oral  work.  Let  the  children  make  problems  for  the  class  to 
solve.  It  will  be  seen  that  in  this  work  the  child's  attention  is  drawn 
to  the  facts  of  number,  and  not  yet  to  the  processes  of  addition  and 
subtraction. 

34.  How  many  cents  eqnal  a  dime  ?  What  else  are 
dimes  called  ?     Ans.   Ten-cent  pieces. 

35.  How  many  cents  eqnal  two  dimes  ?  Tliree  dimes  ? 
Four  dimes  ? 

36.  If  you  had  20  cents,  how  many  dimes  would  you 
have  ? 

37.  If  you  had  30  cents,  how  many  dimes  would  you 
have  ? 

38.  40  cents  equal  how  many  dimes  ? 

39.  Which  is  the  more  money,  31  cents  or  three  dimes  ? 
How  much  more? 

40.  If  you  had  10  cents  and  your  father  gave  you  10 
cents  more,  liow  many  cents  would  you  have  ?  How 
many  dimes  would  they  equal  ? 

41.  If  3'ou  had  20  cents  and  your  father  gave  you  10 
cents  more,  how  much  money  would  you  have  ? 

42.  If  you  had  30  cents  and  your  brother  gave  you  10 
cents  more,  how  much  money  would  you  have  ? 

43.  If  you  had  20  cents  and  gave  away  10  cents,  how 
many  cents  would  you  have  left  ? 

44.  If  you  had  30  cents  and  lost  10  cents,  how  mucli 
money  would  you  have  ? 

45.  If  you  had  10  cents  and  your  mother  gave  you  a 
ten-cent  piece,  how  much  money  would  you  have  ? 

HORN.    ARITH.  2 


18  TENS 

46.  If  you  liad  a  dime  and  your  mother  gave  you  10 
cents,  liow  many  cents'  worth  of  apples  could  you  buy 
with  your  money  ? 

47.  How  many  tens  make  20  ?  Point  them  out  in  the 
table. 

48.  Show  how  many  tens  make  30.   40.    60.   80.  50.   90. 

49.  Sixty  means  six  tens  ;  what  does  seventy  mean  ? 
Eighty  ?     Ninety  ? 

50.  When  we  mean  three  tens,  we  do  not  say  threety  ; 
wliat  do  we  say  ? 

51.  How  do  we  express  two  tens?  Four  tens  ?  Five 
tens  ?     Ten  tens  ? 

52.  Can  }Vou  find  out  how  many  tens  make  50  without 
counting  them  ? 

53.  If  you  have  10  cents  and  your  brother  has  11  cents, 
how  many  more  has  he  than  you  have  ? 

54.  If  you  have  10  cents  and  your  sister  has  2  cents 
more  than  you,  how  many  cents  has  she  ? 

55.  10  cents  and  8  cents  equal  how  many  cents  ? 

56.  One  dime  and  5  cents  equal  how  many  cents  ? 

57.  One  dime  and  7  cents  equal  how  many  cents  ? 

58.  One  dime  and  6  cents  equal  how  many  cents  ? 

59.  One  dime  and  8  cents  equal  how  many  cents  ? 

60.  Find  10  in  the  table,  add  4,  and  point  out  the  num- 
ber which  is  the  answer.     In  the  same  way  add  9  to  10. 

61.  Add  7  to  10.  Add  6  to  10.  5  to  10.  3  to  10.  8 
to  10. 

62.  10  +  2  =  ?     10  +  9  =  ?     10  +  4  =  ? 

63.  Find  20  in  the  table,  add  3,  and  point  out  the  num- 
ber which  is  the  answer. 


TENS  19 

64.  In  the  same  way  add  6  to  20.     Add  4  to  20.     Add 
7  to  20.     Add  9  to  20. 

65.  20  +  5  =  ?     20  +  2  =  ?     20  +  8  ?     20  +  4  =  ? 

66.  If  you   have   two   dimes   and   one  cent,  how  many 
cents  in  money  have  you  ? 

67.  If  you  have  2  dimes  and  three  cents,  how  much 
money  have  you  ? 

How  many  cents  are  equal  to : 

68.  2  dimes  and  o  cents  ? 

69.  2  dimes  and  4  cents  ? 

70.  20  cents  hicking  1  cent  ? 

71.  20  cents  lacking  2  cents  ? 

72.  2  dimes  and  7  cents  ? 

73.  2  dimes  and  9  cents  ? 

74.  20  cents  lacking  3  cents  ? 

75.  20  cents  lacking  4  cents  ? 

76.  Find  30,  add  4,  and  point  out  the  answer.     Add  3 
to  30  in  the  same  way. 

77.  Add  5  to  30.    7  to  30.     2  to  30.    8  to  30.    6  to  30. 
9  to  30. 

78.  Three  dimes  =  how  many  cents  ? 

79.  If  you  have  3  dimes  and  2  cents,  how  many  cents 
in  money  have  you  ? 

80.  If  you  have  3  dimes  and  4  cents,  how  much  money 
have  you  ? 

How  many  cents  are  equal  to: 

81.  3  dimes  and  3  cents  ?  85.  3  dimes  and  5  cents  ? 

82.  3  dimes  and  7  cents  ?  86.  3  dimes  and  9  cents  ? 

83.  30  cents  less  1  cent  =  ?      87.  30  cents  less  4  cents  =  ? 

84.  30  cents  less  3  cents  =  ?     88.  30  cents  less  2  cents  =  V 


20  TENS 

89.  11  cents  equal  how  many  dimes  and  how  many  cents 
over  ? 

90.  How  much  more  than  a  dime  are  13  cents  ?  15  cents  ? 
17  cents?  14  cents?  19  cents? 

91.  How  mucli  more  than  2  dimes  are  21  cents  ?  24  cents  ? 
22  cents  ? 

92.  How  much  more  than  3  dimes  are  33  cents?  31  cents? 
37  cents? 

93.  If  you  buy  something  for  9  cents  and  give  the  clerk 
a  dime,  how  much  change  ought  you  to  have? 

94.  If  you  buy  something  which  costs  29  cents  and  give 
the  clerk  3  dimes,  how  much  change  should  you  have  ? 

95.  If  you  buy  something  that  costs  18  cents  and  give 
the  clerk  2  dimes,  how  much  change  ought  you  to  have  ? 

96.  If  James  had  3  cents  more  he  would  have  10  cents. 
How  much  money  has  he  ? 

97.  John  has  a  dime,  a  nickel,  and  2  cents.  How  much 
money  has  he  ?  Walter  has  2  dimes.  How  much  more 
has  he  than  John  ? 

98.  Henry  has  2  dimes,  a  nickel,  and  3  cents.  How 
much  more  money  must  he  get  to  have  30  cents  ? 

99.  Mary  has  3  dimes.  If  her  mother  should  give  her 
9  cents,  how  much  money  would  she  have?  How  much 
more  must  she  get  to  buy  something  worth  41  cents  ? 

100.  John  has  3  dimes,  and  James  has  7  cents.  How 
many  cents  liave  they  both  ? 

101.  Find  40  in  the  table,  add  5,  and  point  out  the 
answer. 

102.  Add  8  to  20.  Ada  4  to  20.  Add  3  to  30.  Add 
7  to  30.     Add  2  to  40. 


TENS  21 

103.  Add  8  to  40.  Add  8  to  50.  Add  8  to  50.  Add 
4  to  60.     Add  6  to  60. 

104.  Add  3  to  70.  Add  9  to  70.  Add  5  to  80.  Add 
7  to  80.     Add  1  to  90. 

Show  the  children  how  older  people  write  numbers  when  they  add 
them,  and  let  them  furnish  numbers  for  many  exatnples  similar  to  the 
following.  It  will  be  observed  that  of  the  numbers  combined,  one  is  a 
multiple  of  10,  and  the  other  a  number  less  than  10. 

105. 


A7IS. 


To  10 

To  10 

To  10 

To  20 

add   3 

add    5 

add    7 

add      ■:: 

18  A71S. 

A71S. 

Ans. 

^ 

To  20 

To  80 

To  40 

To  50 

add    6 

add    3 

add    2 

add   4 

Ans. 

Ans. 

Ans. 

- 

To  60 

To  70 

To  80 

To  90 

add    3 

add    6 

add    2 

add   5 

Ans. 

Ans. 

Ans. 

Ans. 


Ans. 

106.  If  you  divide  20  cents  between  two  girls  so  that 
each  gets  the  same,  how  much  will  each  get  ?  How  much 
is  one  half  of  20? 

107.  How  many  dimes  are  one  half  of  4  dimes  ? 

108.  If  you  divide  40  cents  equally  between  two  boys, 
how  much  will  each  get? 

109.  Three  girls  have  10  cents  apiece.  How  much 
have  they  all  together  ? 

110.  Mary  had  20  cents,  and  Kate  had  10  cents.  How 
much  did  they  both  have  ? 

111.  Anna  had  3  dimes,  and  Lucy  had  10  cents.  How 
many  cents'  worth  of  oranges  could  they  buy  with  all  their 
money  ? 


22  TENS 

112.  Helen  had  40  cents,  and  her  mother  gave  her 
10  cents.  How  much  money  had  she  then  ?  She  bought 
a  doll  for  49  cents.     How  much  money  had  she  left  ? 

113.  How  many  cents  make  a  dollar?  How  many 
cents  make  half  a  dollar? 

114.  Which  is  the  more  money,  53  cents  or  half  a  dol- 
lar ?     How  much  more  ? 

115.  Find  30  in  the  table,  add  10,  and  show  the  answer. 

116.  Add  10  to  50.  Add  10  to  70.  Add  10  to  40. 
Add  10  to  80.     Add  10  to  90. 

117.  To  20         To  40         To  80         To  60  To  50 
add  10       add  10        add  10        add  10         add  10 


118.  Begin  at  10  and  count  by  tens  to  100. 

119.  Fill  out  and  learn  the  following  : 

1  ten    =  6  tens  = 

2  tens  =  7  tens  = 

3  tens  =  8  tens  = 

4  tens  =  9  tens  = 

5  tens  =  10  tens  = 

120.  10,  20,  30,  etc.,  are  called  multiples  of  10.  Begin 
with  10  and  name  the  multiples  of  10  as  far  as  you  can. 

"  Multiple "  is  not  a  difficult  word  for  children  when  it  is  used  in 
its  objective  sense,  as  in  this  case.  The  author's  pupils  used  to  call 
the  multiples  "bright  numbers"  until  experience  showed  the  advan- 
tage of  giving  them  the  true  name. 

121.  What  is  the  first  multiple  of  10  ?     Ans.  10. 

122.  What  is  the  second  multiple  of  10  ? 

123.  Point  out  the  third  multiple  of  10. 

124.  What  is  the  fourth  multiple  of  10  ?  The  seventh 
multiple  of  10  ?     The  tenth  multiple  of  10  ? 


TENS  23 

125.  What  figure  does  each  of  the  multiples  of  10  end 
witii  ? 

126.  30  is  which  multiple  of  10  ?  50  is  which  multiple 
of  10  ?     70  is  which  multiple  of  10  ? 

127.  How  many  tens  in  70  ?     In  60  ?     In  40  ?     In  90  ? 

128.  80  equals  how  many  tens  ?  50  equals  how  many 
tens  ?  70  equals  how  many  tens  ?  100  equals  how  many 
tens  ? 

3.29.  Write  the  table  of  numbers,  setting  them  in 
straight  columns,  and  making  the  multiples  of  10  larger 
and  brighter  than  the  other  numbers. 

Let  the  children  use  colored  crayons  to  make  the  multiples  distinct. 
Give  a  few  of  the  most  expert  pupils  a  certain  space  at  the  board 
where  they  can  spend  some  time  each  day  making  their  tables,  until 
they  consider  them  fit  to  be  presented  as  their  completed  work.  As 
each  one  finishes  his  table,  give  the  space  to  another  child  until  all 
have  written  it.  The  necessities  of  the  acts  of  construction  make  the 
mental  picture  distinct. 

130.  3  tens  +  4  =  ?   7  tens  +  7  =  ?   8  tens  +  6  =  ? 
9  tens  +  5  =  ?   3  tens  +  2  =  ?   4  tens  +  8  =  ? 

Give  occasionally  chart  exercises  similar  to  the  following : 
Point  out  5,  add  4,  add  2,  take  away  1,  add  7,  take  away  3,  add  5, 
subtract  1,  etc.  Have  the  children  make  the  combinations  without 
counting  as  soon  as  possible.  Encourage  them  to  recite  without  look- 
ing at  the  number  table,  but  do  not  allow  guessing.  Require  them 
to  go  back  to  the  number  table  whenever  they  show  indefinite  ideas 
of  numerical  distances. 

131.  Cover  up  the  last  two  numbers  of  the  first  ten  in 
your  number  table.  How  many  numbers  of  that  column 
are  left  in  sight  ? 

132.  If  we  cover  the  last  two  numbers  of  the  first 
twenty,  how  many  of  the  twenty  are  left? 


24  TENS 

133.  Take  4  from  30  in  the  same  way  and  show  how 
many  are  left. 

134.  Take  2  from  70.     3  from  80.     4  from  90.     3  from 
100. 

135.  Which  is  more,  2  tens  or  19  ?     How  much  more  ? 

136.  Which  is  more,  3  tens  or  28  ?     How  much  more? 

137.  4  tens  are  how  many  more  than  39  ?     Than  37  ? 
Than  33  ?     Than  31  ? 

138.  62  is  how  many  more  than  6  tens? 

139.  83  is  how  many  more  than  8  tens  ? 

140.  What  number  is  2  more  than  28?     2  more  than 
38  ?     Than  48  ?     Than  78  ? 

141.  What  number  is  3  more  than  10?     3  more  than 
20  ?     Than  30  ?     Than  80  ? 

142.  What  number  is  3  less  than  10  ?     3  less  than  40  ? 
Than  50?     Than  60?     Than  80? 

143.  AVhat  number  is  4   more  than  3  tens?      4  more 
than  6  tens  ?    Than  5  tens  ?    Than  7  tens  ?    Than  9  tens  ? 

144.  What  must  be  added  to  79  to  equal  8  tens? 

145.  How  many  must  be  added  to  48  to  equal  5  tens  ? 

146.  How  many  must  be  added  to  7  tens  to  make  72  ? 

147.  How  many  must  be  added  to  3  tens  to  make  35  ? 

148.  How  many  must  be  taken  from  4  tens  to  leave  38  ? 

149.  How  many  must  be  taken  from  5  tens  to  leave  47  ? 

150.  How  many  must  be  subtracted  from  95  to  leave 
9  tens? 

151.  How  many  must  be  subtracted  from  28  to  leave 
2  tens  ? 


TENS  25 

152.  John  had  40  cents  and  lost  2  cents.  How  many 
had  he  left  ? 

153.  Mary  has  10  cents.  Anna  has  3  times  as  many. 
How  man}^  has  Anna  ? 

Let  the  children  make  story  problems  like  the  preceding. 

154.  5  +  4  =  ?     15  +  4  =  !     25  +  4  =  -^     35  +  4  =  ? 

^^  +  4  =  •/     ()5  +  4  =  ?      75  +  4  =  ?     85  +  4  =  ? 

155.  Add  3  to  several  numbers  that  end  in  5,  and  point 
out  the  answers. 

156.  Add:    16      26      36      46      bQ      m      76      86      96 

33333        3   '333 

Will  not  some  of  the  pupils  find  out  for  themselves  that  the  addi- 
tions can  be  performed  easily  by  adding  the  units  and  bringing  down 
the  tens?     This  should  be  shown  to  all. 

157.  Add  4  to  several  numbers  that  end  in  2.  Write 
the  numbers  as  grown  people  write  large  numbers  when 
they  add  them. 

158.  Add  5  to  some  numbers  that  end  in  3. 

159.  Add  2  to  numbers  that  end  in  7.  Add  4  to  num- 
bers that  end  in  1. 

Give  oral  as  well  as  written  work  on  these  combinations  until  they 
are  mastered  and  the  cliildren  no  longer  count. 

Observe  that  the  sum  of  the  units  given  in  this  work  is  less  than  10. 

160.  Add:    46      33      23      51      71      84      62      m      41 

161.  If  3^ou  had  a  figure  made  of  13  squares  and  should 
add  5  squares,  how  many  squares  would  there  be  in  the 
figure  ? 

If  the  children  are  uncertain,  let  the  squares  be  placed,  not  otherwise. 

162.  li  you  had  36  cents  and  earned  3  cents,  how  many 
would  vou  have  ? 


26 


TENS 


163.  Mary  had  23  nuts  and  picked  up  4  nuts;  how  many 
had  slie  then  ? 

Let  the  children  make  story  problems. 

164.  Take  2  from  7.      2  from  17.     2  from  27.     2  from 
37.      2  from  47.      2  from  57. 

165.  Take  2  from  all  the  numbers  on  the  number  table 
that  end  in  8.      Point  out  the  answers. 

166.  In  the  same  way  take  2  from  all  the  numbers  on 
the  number  table  that  end  in  5. 

167      From  39  Show  that  instead  of  thinking  of  the  whole 

,         ^y      ^9  we  can  take  2  from  9  separately,  and  bring 

LaKe    —i     1        ii    .1  i 

down  the  o  tens. 


59 

29 

49 

79 

99 

69 

89 

2 

9 

2 

2 

2 

2 

2 

168.  Write  some  numbers  that  end  in   6,   and  take  3 
from  each  of  them. 

169.  Take  3  from  numbers  that  end  in  9. 

170.  Take  4  from  numbers  that  end  in  8. 

171.  Take  4  from  numbers  that  end  in  6. 

172.  Take  5  from  numbers  that  end  in  9. 

173.  Take  5  from  numbers  that  end  in  7. 

174.  Take  8  from  numbers  that  end  in  9. 

175.  From    27      38      46      88      96      84 

take      2        3        4        3        5        3 


67 

5 


78 
4 


89 
3 


176.  If  you  had  })laced  27  squares  in  a  figure  and  should 
take  away  5  squares,  how  many  would  be  left  ? 

177.  If  there  were  38  squares  in  a  figure  and  you  took 
away  4  squares,  how  many  would  be  left  ? 

178.  John  had  17  marbles  and  lost  4  marbles.     How 
many  were  left  ? 


TENS  27 

179.  Mr.  Smith  earns  28  dollars  a  week  and  spends  7 
dollars  for  board.     How  much  has  he  left  ? 

180.  Mary  had  89  cents  and  spent  6  cents.  How  many 
had  she  left  ? 

Call  for  similar  number  stories. 

181.  Point  out  G,  add  10,  and  point  out  the  answer. 
Add  10  to  26.  To  36.  Go  on  adding  tens  until  you 
reach  96. 

182.  In  the  same  way  add  tens  to  5  until  you  reach  95. 

183.  Mary  may  name  a  small  number  and  the  others 
may  add  tens  to  it. 

It  is  to  be  hoped  that  some  of  the  children  will  go  beyond  the 
limits  of  the  100,  and  that  the  others  will  readily  follow. 

184.  To  35    64    86    48    57    33    72 
add  10    10    10    10    10    10    TO 

Call  attention  to  the  convenience  of  the  plan  of  adding  the  columns 
of  units  and  tens  separately,  and  let  the  children  prove  by  trial  that  it 
gives  the  same  result  as  reckoning  on  the  number  table. 

185.  If  you  had  13  squares  in  a  figure  and  added  10 
squares,  how  many  squares  would  there  be  in  the  figure  ? 

186.  27  squares  and  10  squares  =  how  many  squares  ? 

187.  If  you  had  24  cents  and  earned  10  cents,  how  much 
money  would  you  have  ? 

188.  10  cents  added  to  a  nickel  =  how  many  cents  ? 

189.  10  cents  added  to  a  (juarter  of  a  dollar  =  how 
many  cents  ? 

190.  Point  out  48,  and  subtract  10.  Take  10  from  38. 
From  28.     From  18. 

191.  Subtract  all  the  tens  you  can  from  93.  From  97. 
From  95.     From  91.     From  94. 


28  .  TENS 

192.  John  may  name  a  number,  and  the  others  may 
subtract  from  it  as  many  tens  as  they  can. 

193.  From   87         79         64         21         47         86         73 

take   10         10         10         10         10         10         10 

194.  If  there  were  19  squares  in  a  figure,  and  we  took 
away  10  squares,  hoAV  many  Avould  be  left  ? 

195.  38  squares  lacking  10  squares  =  how  many  squares  ? 

196.  If  Anna  had  25  cents,  and  h)st  10  cents,  how  many 
would  she  have  left  ? 

197.  If  William's  father  had  35  dollars,  and  spent  10 
dollars  for  William's  suit  of  clothes,  how  many  dollars 
would  he  have  left? 

198.  48  chickens  were  in  a  coop,  and  10  chickens  were 
sold.     How  many  were  left  ? 

199.  Add  2  tens  to  21.  3  tens  to  42.  4  tens  to  54. 
3  tens  to  26.     4  tens  to  38. 

200.  Name  a  number  smaller  than  50,  and  add  5  tens 
to  it. 

201.  Name  a  number  smaller  than  30,  and  add  7  tens 
to  it. 

202.  Name  a  number  smaller  than  40,  and  add  4  tens 
to  it. 

203.  Name  a  number  smaller  than  20,  and  add  5  tens 
to  it. 

204.  Add:  68  43  57  29  19  36 

30  40  30  40  80  50 

205.  Subtract  2  tens  from  47.  3  tens  from  83.  5  tens 
from  79. 

206.  Name  a  number  larger  than  50,  and  subtract  4  tens 
from  it. 


TENS  29 

207.    Name  a  number  larger  than  80,  and  subtract  7  tens 
from  it. 


208 


Name  a  number  larger  than  40,  and  subtract  2  tens 


from  it. 

209.  Name  a  number  larger  than  70,  and  subtract  5  tens 
from  it. 

210.  From   79         66        85         73         89         98         54 

take  30         20         50         40         60         80         30 

211.  Find  17,  add  10,  add  20  to  the  result,  subtract  10, 
add  30,  add  20,  subtract  30,  add  10,  subtract  20. 

212.  Find  93,  subtract  10,  subtract  20,  subtract  10,  add 
20,  subtract  20,  add  10,  subtract  20. 

Give  similar  chart  exercises  frequently. 

213.  Find  25  and  add  10,  and  then  1. 

214.  Find  23  and  add  11.     Add  11  to  46.     Add  11  to 

75.     To  81.     To  58. 

Call  the  children's  attention  to  the  relative  position  of  numbers  in 
the  diagram  whose  difference  is  11.  Do  not  let  them  count  11,  but 
add  10  and  then  1. 

215.  Begin  with  11  and  add  elevens  until  you  reach  99. 

216.  Find  a  number  greater  than  12  and  less  than  16, 
and  add  2  elevens  to  it. 

217.  Find  a  number  greater  than  23  and  less  than  26, 
and  add  2  elevens  to  it. 

218.  Add  2  elevens  to  each  of  the  numbers  that  are 
between  34  and  37. 

219.  Think  of  12,  and  without  counting,  add  11  to  it. 
Keep  on  adding  elevens  until  you  reach  100. 

220.  Think  of  a  number  greater  than  21  and  less  than 
25,  and  add  2  elevens  to  it, 


30  TENS 

221.  Add  23   75   68   74   26   33   48   87 

11   11   11   11   11   11   11   11 

222.  Point  out  on  the  number  table  the  number  that 
means  2  elevens,  and  add  elevens  to  it  until  you  reach  41. 
How  many  elevens  did  you  add? 

223.  Add   36  74  37  55  41  34 

22  22  22  22  22  22 

224.  If  James  had  11  cents  and  John  had  as  many 
more,  how  many  did  they  both  have?  How  many  dimes, 
and  how  many  cents  over  would  their  money  equal? 

225.  If  James  had  24  cents  and  John  had  11  cents  more 
than  James,  how  many  did  John  liave  ?  How  many  dimes 
and  how  many  cents? 

226.  Mr.  Smith  paid  75  dollars  for  a  horse  and  11  dol- 
lars for  a  harness.      How  much  did  they  both  cost  ? 

Call  for  story  problems,  using  the  number  11. 

227.  Find  59  and  take  away  11.  Take  11  from  83. 
From  75.     From  84.     From  97.      From  48. 

228.  From  68  43  84  65  48  76 

take  11  11  11  11  11  11 

From  35  66  84  29  77  98 

take  22  22  22  22  22  22 

229.  A  man  had  33  dollars  and  lost  11  dollars.  How 
much  had  he  left? 

230.  24  apples  less  11  apples  =  how  many  apples  ? 

231.  36  squares  less  11  squares  =  how  many  squares  ? 

232.  Find  25  on  the  chart,  add  10,  and  then  2. 

233.  Add  12  to  24.  Keep  on  adding  twelves  until  you 
reach  96.     Remember  that  12  means  10  and  2. 


TENS  31 

234.  Think  of  a  number  greater  than  40  and  less  than 
48,  and  add  12  to  it. 

235.  Add  12  to  each  of  the  numbers  greater  than  20 
and  less  than  26. 

236.  Think  of  a  number  between  31  and  34,  and  add  3 
twelves  to  it. 

237.  Think  of  a  number  l)etween  63  and  66,  and  add  2 
twelves  to  it. 

238.  Add:  82      47      63      75      84      5()      24      66      87 

12      12      12      12      12       12      12      12      12 

239.  Write  12  under  76  and  add  them.  Write  12 
under  44  and  add  them. 

240.  If  you  had  12  cents  and  received  12  cents,  how 
many  cents  would  you  have  ?  If  you  gained  another  12 
cents,  how  many  cents  would  you  have  ?  How  many 
dimes,  and  how  many  cents  over  ? 

241.  If  you  had  15  cents  and  gained  12  cents,  how 
many  cents  would  you  have  ? 

242.  Point  out  on  the  number  table  the  numbers  that 
mean  2  twelves.  3  twelves.  4  twelves.  5  twelves.  6 
twelves.     7  twelves. 

243.  Find  the  number  that  means  3  twelves ;  write  it 
under  42  and  add. 

244.  Write  under  21  the  number  that  means  4  twelves, 
and  add. 

245.  Add  15  to  the  number  that  means  2  twelves. 

246.  Find  48,  take  12  from  it,  and  point  out  the  number 
that  is  left.     Keep  on  taking  twelves  until  nothing  is  left. 

247.  Find  59  and  take  twelves  from  it  until  11  remains. 


32  TENS 

248.  From     88  96  34  78  25  57 
take       12           12           12           12           12  12 

249.  Find  23  on  the  chart  and  add  13  to  it.  Add  13 
to  13.     To  53. 

250.  Add  14  to  23.  Add  14  to  33.     To  53.     To  73. 

251.  Add  15  to  24.  To  44.     To  74.     To  34. 

252.  Add  16  to  21.  To  31.     To  51.     To  81.     To  61. 

253.  Add  17  to  22.  To  82.     To  32.     To  52.       . 

254.  Add  22     31     84     24     25     85     81      33     24     51 

1717131313131414     15     15 

255.  From  77     86     47     62     48     55     78     69     58     53 
take     24     35     33     30     26     43     24     37     27     21 

256.  How  many  chiklren  are  there  in  your  class?  If 
they  were  pUiced  in  groups  of  10,  how  many  groups  would 
there  l)e,  and  how  many  children  over  ? 

257.  12  cents  =  how  many  dimes  and  cents  ? 

258.  18  cents  =  how  many  dimes  and  cents  ? 

259.  33  cents  =  liow  many  dimes  and  cents  ? 

260.  Write  32  cents  in  dimes  and  cents. 

261.  Write  in  dimes  and  cents  45  cents.  75  cents.  24 
cents.     38  cents. 

262.  How  many  are  3  tens  and  5  ?  7  tens  and  6  ?  2 
tens  and  7  ?     4  tens  and  3 

263.  How  many  ones  in  4  ?     How  many  ones  in  7  ? 

264.  Sometimes  the  word  "unit"  is  used  to  mean  one. 
How  many  units  in  6?     In  9? 

265.  11  means  1  ten  and  1  unit.      What  does  12  mean? 

Ani<.  1  ten  and  2  units. 


TENS  33 

Take  numbers  consisting  of  tens  and  units,  as  42,  and  lead  the 
children  to  see  that  the  figure  4  stands  for  4  tens  (which  they  may  show 
by  the  number  table,  or  by  columns  of  squares),  and  that  the  figure  2 
stands  for  the  2  remaining  units. 

266.  What  does  17  mean?  21?  32?  64?  57?  63?  89? 

267.  Separate  37  into  tens  and  units. 

268.  Separate  into  tens  and  units  48,  57,  65,  39,  82,  68, 
95,  24. 

269.  How  do  you  write  3  tens  and  7  units  together? 
5  tens  and  2  units  ?     7  tens  and  8  units  ? 

270.  In  the  number  25,  which  figure  stands  for  tens  ? 
Which  figure  stands  for  units? 

271.  In  the  number  68,  which  figure  is  in  the  tens' 
place,  and  which  is  in  tlie  units'  phice  ? 

272.  Write  a  number  whicli  has  4  in  the  tens'  phice  and 
7  in  the  units'  place. 

273.  What  number  has  5  in  the  tens'  place,  and  3  in  the 
units'  place  ? 

274.  Write  some  numbers  of  two  places,  and  tell  what 
figures  you  put  in  the  tens'  places,  and  what  figure  in 
the  units'  places. 

275.  Write  a  number  which  has  9  in  the  units'  place. 
Can  you  write  a  number  which  has  10  in  the  units'  place? 

276.  When  we  mean  20  and  9,  we  write  29.  What  do 
we  write  when  we  mean  20  and  10  ?  20  and  11  ?  20 
and  12  ? 

4  units  and  2  units  =  liow  many  units  ? 

277.  To  3-1        (3  tens  and  3  tens      =  how  many  tens  ? 
add  64        How  many  tens  and  how  many  units  in 

the  answ^er  ? 

HORN.     ARITH.  3 


34  i^^^!^ 

278.  Write  52  under^  46  and  add  them.  Why  is  it 
best  to  write  units  under  units,  and  tens  under  tens,  when 
we  add  numbers  ? 

279.  Add  76  and  22.  Add  23  and  34.  Add  25  and  42. 
Add  31  and  63.  Add  34  and  25.  Add  11  and  78.  Show 
how  many  tens  and  how  many  units  in  each  answer. 

280.  Write  22  under  48,  and  subtract  the  smaller  num- 
ber from  the  greater. 

281.  From  39  take  11.  From  27  take  15.  From  78 
take  22.     From  66  take  21.      From  83  take  62. 

282.  What  is  the  largest  number  that  is  written  with 
one  figure  ? 

283.  What  is  the  smallest  number  that  is  written  with 
two  figures? 

284.  What  is  the  largest  number  that  can  be  expressed 
by  two  figures  ? 

285.  What  is  the  smallest  number  that  can  be  expressed 
by  three  figures  ? 

It  should  be  explained  that  there  were  people  living  long  ago,  called 
Romans,  who  expressed  numbers  by  letters  instead  of  figures,  and 
that  sometimes  we  still  use  then'  notation. 

286.  X  stands  for  10  in  Roman  notation.  Find  the 
lOtli  chapter  in  this  book,  and  tell  on  what  page  it  begins. 
Make  a  10  like  that  on  the  clock. 

28V.  Since  X  stands  for  10,  what  do  two  X's  stand  for  ? 
What  do  three  X's  stand  for  ? 

288.    Write  in  Roman  notation,  10,  20,  30. 


CHAPTER    III 

TWOS 

Even  Numbers,  Foot  and  Inch,  Halves,  Quart  and  Pint, 
Horizontal  Line,  Triangle,  Thirds,  Fourths,  Verti- 
cal Line,  Roman  Numeral  I 

NUMBER  TABLE 

1  11  21  31  41  51  61  71  81  91 

2  12  22  32  42  52  62  72  82  92 

3  13  23  33  43  53  63  73  83  93 

4  14  24  34  44  54  64  74  84  94 

5  15  25  35  45  55  6,"^  15  85  95 

6  16  26  36  46  56  66  16  86  96 
1  17  27  37  47  57  67  77  87  97 

8  18  28  38  48  58  68  78  88  98 

9  19  29  39  49  59  69  79  89  99 

10    20    30    40    50   60   70   80   90   100 

1.  Begin  with  2,  and  connt  by  twos  to  40,  pointing  ont 
the  nnmbers  on  the  number  table.  Count  without  the 
number  table. 


35 


36  TWOS 

2.  These  numbers  that  you  have  been  giving,  2,  4,  6,  8, 
and  so  on,  are  called  Even  Numbers.  Name  all  the  even 
numbers  in  the  1st  ten.      In  the  2d  ten.     In  the  3d  ten. 

Call  attention  to  the  endings  of  the  even  numbers. 

3.  Name  all  the  even  numbers  in  the  10th  ten.  In  the 
9th  ten.  Tell  how  many  tens  and  how  many  units  in  each 
even  number  in  the  9th  ten. 

4.  What  is  the  smallest  even  number  that  you  can 
think  of? 

5.  Name  all  the  even  numbers  that  end  in  2.  In  4. 
In  8.  In  0.  What  other  figure  may  an  even  number 
have  in  its  units'  place  besides  2,  4,  8  or  0  ? 

6.  What  is  the  smallest  even  number  in  the  3d  ten  ? 
In  the  7th  ten?     In  the  10th  ten? 

7.  What  is  the  largest  even  number  in  the  4th  ten? 
In  the  6th  ten?     In  the  8th  ten  ? 

8.  Think  of  the  largest  even  number  that  is  less  than 
29  and  write  it. 

9.  What  is  the  next  even  number  after  34?  Before 
34? 

10.  Find  the  third  even  number.  The  fifth  even 
number. 

11.  What  even  number  comes  just  before  49?  How 
man}^  tens  and  how  many  units  in  it? 

A  device  for  leading  children  to  recognize  even  numbers  is  to  write 
an  even  number  out  of  the  children's  sight,  and  then  let  them  guess 
what  it  is,  giving  them  a  clew^  as  "  It  is  in  the  3d  ten  "  or  "  It  ends 
with  4,"  or  "  It  is  between  91  and  90,"  •'  It  is  larger  than  26,  but  not 
so  large  as  38,"  Sontetimes  allow  the  children  to  write  the  hidden 
number. 


TWOS  37 

12.    Write  the  tirst  30  numbers,  marking  them  off  into 
groups  of  two,  as  follows  : 


13.  How  many  groups  of  2  equal  8? 
How  many  equal  12?  18?  14?  20?  16? 
24?  30? 

14.  How  many  ones  in  3  twos?  In  5 
twos?  In  8  twos?  In  6  twos?  In  9 
twos?     In  12  twos?     In  7  twos? 

15.  Name  some  even  numbers,  and 
tell  how  many  twos  they  equal. 


1 

11 

21 

2 

12 

22 

3 

13 

23 

4 

14 

24 

5 

15 

25 

6 

16 

26 

7 

17 

27 

8 

18 

28 

9 

19 

29 

10 

20 

30 

16.  Fill  out  ai 

1  two 

= 

2  twos 

:^ 

3  twos 



4  twos 

= 

17.  54  -h 

9  —  > 

5  twos  =  9  twos  = 

6  twos  =  10  twos  = 

7  twos  =  11  twos  = 

8  twos  =  12  twos  = 

84  +  2  =  ?        94  +  2  =  ?        74  4-  2  =  ? 

18.  Find  42  in  the  number  table,  add  2  twos,  and  point 
out  the  answer. 

19.  Find  34,  and  add  3  twos.     Add  4  twos  to  50.     Add 
5  twos  to  20. 

20.  Add  2  twos  to  18.     Add  3  twos  to  28.     Add  4 
twos  to  36. 

21.  Find  76,  add  6  twos,  and  show  the  answer.     Add  7 
twos  to  64.     Add  8  twos  to  24.     Add  9  twos  to  20. 

22.  Point  out  even  numbers  and  add  some  twos  to  them. 

23.  How  many  twos  must  be  added  to  12  to  equal  18? 
To  equal  22?     To  equal  16? 


38  TWOS 

24.  How  many  twos  must  be  added  to  14  to  equal  20? 
To  equal  24?     To  equal  18? 

25.  If  you  had  18  cents  and  gained  2  cents,  how  many 
would  you  have?     How  many  dimes  Avould  it  equal? 

26.  If  you  had  16  cents,  and  your  mother  gave  you 
2  cents,  and  your  father  gave  you  2  cents,  how  many  cents 
would  you  have? 

27.  If  you  had  18  cents,  and  3  people  each  gave  you 
2  cents,  how  many  cents  would  you  have? 

28.  If  you  had  10  cents,  and  5  people  each  gave  you 
2  cents,  how  many  cents  would  you  have? 

29.  Add  : 

54     66     36      32      86      64     56 
12      22     42      54      12      32      22 


42 

52 

74 

22 

34 

72 

84 

34 

26 

14 

46 

24 

22 

12 

30.  Beginning  at  20,  count  backwards  by  twos. 

31.  2  from  36  leaves  how  many?     2  from  48  leaves  how 
many?     2  from  96?     18-2:=?     28-2=?     78-2  =  ? 

Give   questions   in    subtraction    similar    to   the   addition    drill    in 
Ex.  18-24. 

32.  If  you  had  36  cents  and  lost  2  cents,  how  many 
would  you  have  left? 

33.  If  you  liad  24  cents  and  lost  2  cents,  how  many 
would  you  have  left? 

34.  If   you  had    2    dimes    and    bought   something   for 
2  cents,  how  much  money  would  you  have  left? 

35.  If   you  had  a  dollar  and  lost  2  cents,  how  much 
would  you  have  left? 


TWOS  39 

36.  If  20  children  were  at  a  party  and  2  went  home, 
how  many  would  be  left?  How  many  would  be  left  when 
2  more  went  home  ?     When  2  more  went  home  ? 

37.  If  there  were  22  children  belonging  in  your  class 
and  2  were  absent,  how  many  would  be  present?  If  2  of 
those  were  dismissed,  how  many  would  be  left  ? 

38.  Find  20  and  show  how  many  twos  must  be  sub- 
tracted from  it  that  l-i  may  be  left. 

39.  How  many  twos  must  be  subtracted  from  22  to 
leave  16  ?     To  leave  12  ? 

40.  Beginning  at  30,  count  backwards  2  twos.  What 
number  did  you  reach  ? 

41.  Beginning  at  40,  count  backwards  by  twos  until 
you  reach  32.      How  many  twos  did  you  count  off? 

42.  How  many  twos  must  be  taken  from  36  to  leave 
28  ?     24  ?     20  ? 

43.  If  you  had  40  cents  and  gave  2  cents  to  each  of  4 
boys,  how  many  cents  would  you  have  left  ? 

44.  From     68     76     86     38     44     56     32     6Q     54     86 

take     26     34     42     12     22     34     22     42     24     36 

45.  If  you  had  40  cents  and  each  of  5  boys  gave  you 
2  cents,  how  many  cents  would  you  have  ?  How  many 
dimes  would  they  equal  ? 

46.  Draw  with  a  foot  rule  a  line  a  foot  long,  and  mark 
off  the  inches.     How  many  inches  make  a  foot  ? 

■  47.  Draw  a  line  10  inches  long,  marking  tlie  inches. 
A  10-inch  line  lacks  how  many  inches  of  being  as  long  as 
a  line  a  foot  long  ?  Show  how  many  times  a  2-inch  line 
can  be  measured  off  on  a  10-inch  line. 


40  TWOS 

48.  Make  the  line  that  is  a  foot  long,  2  inches  longer. 
How  long  is  it  now  ?  How  much  longer  is  it  than  the 
10 -inch  line  ? 

49.  How  many  times  can  a  2-inch  line  be  measured  off 
on  a  14-inch  line  ? 

50.  Lengthen  the  14-inch  line  2  inches,  and  tell  how 
long  it  is.     How  many  times  will  it  contain  a  2-inch  line? 

51.  Lengthen  the  16-inch  line  2  inches  and  tell  how 
long  it  is,  and  how  many  times  it  contains  a  2-inch  line. 

52.  Lengthen  the  18-inch  line  2  inches  and  tell  how 
long  it  is  and  how  many  times  it  contains  a  2-inch  line. 

53.  16  divided  into  groups  of  2  =  how  many  groups  ? 

54.  24  divided  into  groups  of  2  =  how  many  groups? 

55.  20  -  2  =  ? 

Show  division  as  a  process  of  separating  tlie  larger  number  into 
groups  of  the  less.  Illustrate  by  grouping  objects,  marks,  or  the  num- 
bers of  the  number  table. 

56.  14 -f-    2=?     22-    2='>     12-    2=*^    18--    2=*;^ 
20-1-10=?     40-10=?     70-10=?    30-10=? 

57.  How  many  times  can  a  10-inch  line  be  measured 
off  on  a  20-inch  line  ? 

58.  How  long  is  a  line  which  is  one  half  as  long  as  a 
20-inch  line  ? 

59.  Cut  a  slip  of  paper  12  inches  long,  double  it  to  find 
the  middle,  and  tell  how  many  inches  long  one  half  of  the 
strip  is. 

60.  Measure  off  a  14-inch  line  and  find  how  many 
inches  one  half  of  it  measures. 


TWOS  41 

61.  Fill  out  and  learn  the  following  : 

One  half  of    2  =  ?  i  of  12  =  ? 

One  half  of    4  =  ?  i  of  U  =  ? 

i  of    6  =  ?  -1-  of  IG  =  ? 

1  of    8  =  ?  1  of  18  =  ■? 

-1-  of  10  =  ?  1  of  20  =  ? 

62.  Find  4  in  the  table,  and  point  out  the  number  which 
means  one  half  of  it. 

63.  Show  10  and  the  number  which  is  -J  of  it.  Show  J 
of  12.     Of  16.     Of  20.     Of  14.     Of  18.  ^ 

64.  Mary  has  14  cents,  and  Julia  has  ^  as  many.  How 
many  has  Julia  ? 

65.  How  many  eggs  in  ^  a  dozen  ? 

66.  When  candy  is  10  cents  a  pound,  how  many  cents 
will  |-  a  pound  cost  ? 

67.  AVhen  candy  is  20  cents  a  pound,  how  much  will  I 
a  pound  cost  ? 

68.  How  many  pints  make  a  quart  ? 

Let  the  children  pour  the  water  from  pint  measure  to  quart  meas- 
ure, and  find  out  the  fact  for  themselves. 

69.  How  many  pints  in  2  quarts?  In  4  quarts?  In  li 
quarts  ?  In  7  quarts  ?  In  9  quarts  ?  In  3  quarts  ?  In  8 
quarts?     In  12  quarts?     In  5  quarts? 

70.  If  a  bucket  has  12  joints  of  water  in  it,  how  many 

quarts  of  Avater  can  be  taken  out  of  it  ? 

The  child  who  cannot  imagine  the  process  should  be  allowed  to 
work  it  out  practically. 

71.  How  many  quarts  in  12  pints?  In  18  pints?  In 
14  pints?     In  10  pints? 

72.  A  pint  of  milk  equals  what  part  of  a  quart? 


42  i'Wos 

73.  If  a  quart  of  milk  costs  6  cents,  how  much  does  a 
pint  cost  ? 

74.  If  a  quart  of  vinegar  costs  8  cents,  how  much  does  a 
pint  cost  ? 

75.  If  a  quart  of  molasses  costs  18  cents,  how  much 
does  a  pint  cost  ? 

Teach  the  horizontal  lines. 

76.  Draw  a  horizontal  line  20  inches  long.  Into  how 
many  2-inch  lines  can  it  be  divided  ?  A  12-inch  line 
=  how  many  2-inch  lines  ?  A  16-inch  line  =  how  many 
2-inch  lines?     An  18-inch  line  =  how  many  2-inch  lines? 

77.  Seven  2-inch  lines  =  how  lono"  a  line  ?  P^leven 
2-inch  lines  ?     Twelve  2-inch  lines  ? 

This  should  be  read:  ''2  inches  mul- 


78.    2  inches  x  (>  =  ?     ^-  ,•   i  i     <>     o 

tiplied  l)y  (j  =  r 


?> 


79.  2  inches  x  7  =  ?  2  inches  x  9  —  ? 
2  inches  x  12  =  ?  2  inches  x  10  =  ? 
2  inches  x    4  =  ?                          2  inches  x     8  =  ? 

80.  What  name  is  given  to  numbers  which  equal  any 
number  of  twos  ? 

81.  Name  all  the   even   numbers  between  11  and  19. 
Between  21  and  31. 

82.  Begin  at  18  and  name  all  the  even  numbers  to  84. 

83.  How  many  even  numl)ers  are  tliere  less  tlian  20? 
How  many  less  than  26  ? 

84.  What  even  number  is  3  more  than  31  ?     How  many 
tens  and  how  many  units  in  it? 

85.  How  many  gloves  in  7  pairs  of  gloves  ? 

86.  How  many  shoes  in  9  pairs  of  shoes  ? 

87.  How    many   pints    of    milk    in    8    quarts  ?     In    10 
quarts  ?     lull  quarts  ? 


TWOS 


43 


88.  How  many  slices  can  5  boj^s  wear  at  the  same  time  ? 

89.  12  men  paid  2  dollars  apiece  to  lure  a  sailboat. 
How  much  did  they  all  pay  ? 

90.  11  boys  put  in  2  cents  apiece  to  buy  a  ball.  How 
much  did  they  all  put  in  ? 

91.  How  many  wheels  have  10  bicycles  ? 

92.  T  chickens  have  how  many  feet  ? 

93.  How  much  will  11  two-cent  postage  stamps  cost? 

94.  8  ])airs  of  horses  =  how  many  horses? 

95.  How  many  wings  have  12  Ijirds  ? 

96.  If  you  had  18  a|)plcs,  to  how  many  cliildren  couhl 
you  give  2  apples  apiece  ? 

97.  20  children  are  going  to  march  by  couples.  How 
many  couples  will  there  be?  If  there  were  1(3  children, 
into  how  many  couples  could  they  be  formed  ?  1(3  h-  2  =  ? 
14  -^  2  =  *? 

98.  If  there  were  13  cliildren,  into  how  many  couples 
could  they  be  formed  ?      (Illustrate  if  necessary.) 

99.  Place  inch-squares 


Fig.  1 


as  in  Fig.  1. 

Push  the  squares  apart 
so  as  to  divide  your  fig- 
ure into  halves.  How 
many  square  inches  in 
the  whole  figure?       How  many  in  each  half? 

After  the  children  have  made  the  figures,  it  would  be  well  for  the 
teacher  to  have  them  close  their  books  and  to  give  them  the  questions 
orally,  develoi^ing  ideas  of  forms  and  of  the  ratio  of  their  parts  as  far 
as  the  abilities  of  each  class  allow. 


44 


TWOS 


Fig.  a 


100.  Place  inch -squares  as  in 
Fig.  2.  Divide  the  figure  .into 
halves.  How  many  squares  in  each 
half?  Can  you  put  the  halves 
of  Fig.  2  together  so  as  to  make 
Fig.  i  ? 

101.  Which  is  the  larger,  Fig.  1 
or  Fig.  2  ? 

102.  Place  inch-squares 
as  in  Fig.  3.  Divide  the 
ligure  into  halves,  and 
tell  how  many  square 
inches  in  each  half.  Can 
you  put  the  halves  of 
Fief.  3  toofether  so  as  to 
make  Fig.  2  ? 

103.  Which  is  the  largest.  Fig.  1,  Fig.  2,  or  Fig.  8  ? 
Show  some  horizontal  lines  in  Fig.  3. 

Let  the  children  combine  squares  into  figures  of  their  own  devis- 
ing, divide  the  figures  into  halves,  and  report  upon  them.  Encourage 
symmetrical  forms. 

104.  Cut  an  inch-square  in  two  so  as  to  make  triangles 
like  these. 


Fig. 


105.  Place  the  triangles  as  in  Fig.  4. 
How  many  triangles  would  it  take  to  make 
6  such  figures. 

106.  Place  other  triangles  as  in  Fig.  5. 
How  many  triangles  would  it  take  to  make  9 
figures  like  Fig.  5  ? 


Fig.  4 


Fig.  5 


TWOS 


45 


107.  Place  others  as  in  Fig.  6.  How  many 
triangles  would  you  need  to  make  7  figures 
like  Fig.  6  ? 

108.  Place  others  as  in  Fig.  7.  Wliicli  is 
the  largfest  of  these  fio-ures  ? 


Fig.  7 


Let  the  children  show  the  equality  of  the  figures  by  rearranging 
each  of  them  into  a  square. 


109.  Copy  Fig.  8  by  placing  triangles. 
How  many  triangles  does  it  take?  How 
many  inch  squares  does  it  take  to  make  the 
triangles  used  in  copying  Fig.  8? 

110.  Make  Fig.  9.  Which  is  greater,  Fig. 
8  or  Fig.  9?  How  many  square  inches  in 
each?     How  many  square  inches  in  both? 


Fig.  8 


Fig.  y 


111.  Place  three  triangles  so  as  to  make  other  figures, 
and  show  how  many  square  inches  in  each  of  the  figures. 

112.  How  many  such  triangles  can  you  make  from  one 
inch  square?  From  3  squares?  From  7  squares?  From 
8  squares?     From  9  squares?     From  10  squares? 

113.  12  such  triangles  =  how  many  inch  squares? 

If  the  children  are  uncertain,  let  them  work  out  the  problem  by 
arranging  the  12  triangles  into  inch  squares. 

114.  14  such  triangles  =  how  many  inch  squares?  18 
triangles  =  ?  20  triangles  =  ?  22  triangles  =  ?  24  tri- 
angles =  ? 

115.  10  =  how  many  twos?  2  tens  =  how  many  twos? 
2  tens  =  how  many  units?     10  twos  =  how  many  units? 


46  TWOS 

116.  5  tens  +  2  units  =  how  many  units?  6  tens  +4 
units  =  how  many  units? 

117.  If  Mary  had  3  dimes  and  earned  8  cents,  how 
much  money  woukl  she  have? 

118.  If  John  had  7  dimes  and  a  nickel,  how  much  money 
wouhl  he  have? 

Call  for  story  problems. 

119.  What  is  an  even  number  ? 

120.  Name  an  even  number  smaller  than  20,  and  tell 
how  mau}^  twos  it  equals. 

121.  Into  how  many  parts  is  this  circle 
divided? 

122.  When  anything  is  divided  into  three 
equal  parts,  what  is  eacli  part  called  ? 

Ans.   One  third,  written  i. 

Show  -J  of  the  circle.     Show  J  of  it. 

123.  Draw  a  line  3  inches  long.     Mark  off  -J  of  it. 

124.  Cut  a  strip  of  paper  3  inches  long  and  fold  it  into 
thirds.     Show  -^  of  it.     Show  J  of  it. 

125.  Find  Fig.  8,  Ex.  109,  and  show  how  much  J  of  it 
is.     Show  I  of  it. 

126.  Show  I  of  Fig.  9,  Ex.  110.  Show  i  of  Fig.  2, 
Ex.  100. 

Let  the  children  estimate  |  of  the  length  of  a  book  or  stick  or  some 
convenient  object.  Cut  a  strip  of  paper  as  long  as  the  object,  and 
fold  it  into  thirds  to  test  the  correctness  of  the  estimates. 

Provide  a  tumbler  of  cylindrical  shape,  and  require  a  child  to  bring 
it  to  you  i  full  of  water ;  |  full ;  |  empty ;  i  empty. 


TWOS 


47 


127.  If  you  jjut  ^  of  a  stick  into  the  ground,  how  much 
will  be  above  ground  ? 

128.  The  snow  was  so  deep  that  it  covered  J  of  a  fence 
post.     What  part  of  the  post  was  bare? 

129.  What  is  J  of  6  squares  ?     ^  of  9  squares  =  how 
many  ?      (Illustrate.) 

130.  How  much  is  ^  of  3  cents?     |  of  3  cents  =  how 
many  ? 

131.  Turn  to  the  number  table  and  show  ^  of  3.     ^  of 


o 


f  30. 


6.     i  of  9. 

If  the  children  do  not  see  these  relations  readily,  do  not  let  them 
memorize  the  statements  of  them,  but  postpone  the  subject. 

132.  Place  triangles  as  in  Fig.  10.  How 
many  triangles  does  it  take  ?  How  many 
square  inches  do  they  equal  ?  yig  10 

133.  Place  triangles  as  in  Fig.  11  and  Fig.  12.  Which 
is  the  larger  figure  ?  How  many  triangles  in  both  ?  How 
many  square  inches  do  they  both  equal  ? 


Fig.  11  Fig.  12 

134.  Place  triangles  as  in  Fig.  13,  Fig.  14,  and  Fig.  15. 
Which  is  the  largest  figure  ?  How  many  triangles  in  all  ? 
How  many  square  inches  do  they  all  equal  ? 


Fig.  13 


Fig.  14  Fig.  15 

135.    Place  4  triangles  so  as  to  make  a  figure  different 
from  any  in  the  book. 


48  TWOS 

136.  Fold  a  strip  of  paper  into  4  equal  parts.  What  is 
each  part  called  /     Ans.   One  fourth,  written  -|. 

137.  Draw  a  picture  of  a  pie  cut  into  fourths.  If  ^ 
were  eaten,  how  many  fourths  would  be  left  ? 

138.  Take  a  piece  of  string  and  divide  it  into  fourths. 

139.  If  eight  children  made  a  class,  how  many  children 
would  ^  of  the  class  be  ?     (Illustrate.) 

140.  Pour  water  into  a  glass  until  it  is  ^  full. 

141.  Turn  to  Fig.  10,  Ex.  132,  and  show  ^  of  it.  Show 
|.      Show  1^. 

142.  Show  ^  of  Fig.  11.      Show  I  of  it. 

143.  Find  Fig.  12,  cover  up  ^  of  it,  and  tell  how  many 
fourths  are  in  sight. 

144.  Draw  a  horizontal  line  8  inches  long  and  show  ^ 
of  it.     Show  1^  of  it.     Show  |  of  it. 

145.  A  line  in  this  position  is  called  a  vertical  line. 
Hold  your  pencil  up  to  show  a  vertical  line. 

146.  Draw  a  vertical  line  4  inches  long,  divide  it  into 
fourths,  and  show  -J  of  it.      Show  |  of  it.      Show  |  of  it. 

147.  Turn  to  the  number  table  and  show  ^  of  4.  ^  of 
8.     {  of  40. 

148.  Take  an  inch  square  of  paper  and  fold  it  into 
fourths  shaped  like  those  in  Fig.  13.  Fold  an  inch  square 
into  fourths  of  other  shapes. 

149.  Draw  a  horizontal  line  1  foot  long  and  divide  it 
mto  fourths.     How  many  inches  long  is  ^  of  a  foot  ? 

150.  Draw  a  circle  (by  marking  around  a  coin,  tumbler, 
or  some  circular  object),  cut  it  out  and  fold  it  into  fourths. 
How  many  fourths  in  the  whole  of  anything  ? 


TWOS  49 

151.  Draw  a  horizontal  line  9  inches  long.  Show  J  of 
it.     Show  I  of  it. 

152.  What  is  the  sixth  multiple  of  10  ? 

153.  Write  in  order  in  a  horizontal  line  all  the  muh 
tiples  of  10  which  you  have  learned  from  the  number 
table.  Are  there  any  other  multiples  of  10  ?  If  so,  write 
some. 

154.  Which  will  cost  more,  6  pints  of  milk  or  3  quarts  ? 
Explain. 

155.  6  pints  +  1  quart  —  1  pint  =  how  many  pints  ? 

156.  7  pints  +  1  (|uart  =  how  many  pints  ? 

157.  3  quarts  +  2  pints  =  how  many  quarts  ? 

158.  4  quarts  —  1  pint  =  how  many  pints  ? 

159.  3  dimes  +  5  cents  —  1  cent  =  how  many  cents  ? 

160.  1  foot  -h  3  inches  =  how  many  inches  ? 

161.  Find  some  horizontal  lines  on  this  page. 

162.  What  is  the  largest  even  number  that  can  be 
written  wdth  one  figure  ? 

163.  What  is  the  largest  even  number  that  can  be 
written  with  two  figures  ? 

164.  What  is  the  smallest  even  number  that  can  be 
written  with  two  figures? 

165.  How  many  thumbs  do  ten  boys  have  ? 

166.  How  many  toes  do  two  boys  have  ? 

167.  Which  is  more,  2  times  10  or  10  times  2  ? 

168.  What  does  X  stand  for  in  Roman  notation?  XX? 
XXX? 

169.  I  stands  for  one.  Find  I  on  the  clock.  What 
does  n  mean  on  the  clock  ?     What  does  III  mean? 

HOKX.  Aiam.  — 4 


50  TWOS 

170.  Which  chapter  of  this  book  are  you  studying  ? 
What  is  the  number  of  the  chapter  before  this  ? 

171.  X  H-  I  =  liow  many  ?  '  XX  +  III  = 
X  +  II  =  how  many  ?  XXX  +  I  = 
X  -f  III  =  how  many  ?  XXX  +11  = 

XX  H-      I  =  how  many  ?  XXX  +  1 1 1  = 

172.  Read  XI   from   the   clock.      Read  XII   from   the 
clock. 

173.  Read  XIII.     XXII.     XXIII.     XXXI.     XXXII. 
XXXIII. 

174.  Write  in  Roman  numbers  12,  32,  22,  13,  21,  31, 
23,  33. 

175.  On  which  page  of  this  book  does  the  11th  chapter 
begin  ?     The  13th  chapter  ?     The  12th  chapter  ? 


176. 

Add: 

72 

86 

14    24 

66 

48  • 

32 

24 

12 

74    32 

20 

40 

66 

177. 

From 

88 

66 

38 

54 

76 

84 

take 

12 

24 

26 

22 

36 

44 

CHAPTER   IV 

ADDITION 

Sum,  Yard,  Rectangle,  Thousands,  Gallon,  Perimeter, 
Peck,  Roman  Numerals  V,  L,  and  C 

Do  not  take  up  a  new  combination  of  numbers  until  the  pupils  are 
able  to  give  promptly  those  already  taken.  Exercises  in  form,  in 
simple  fractions  and  measurements,  and  in  Roman  Numerals  are 
introduced  between  the  combinations  not  only  to  illustrate  them,  but 
to  extend  profitably  the  time  during  which  they  are  learned. 

1.  Add: 48         How  many  units  in  the  answer?     How 

Sb     many  tens  ? 

2.  Add:  73         How  many  units  in  the  answer  ?     How 

35     many  tens  ?     Wliat  do  10  tens  make  ? 

3.  Add:  24      76      81      45      38       52       94       82       88 

81      38      27      62       71       53       18       78       25 

Tell  how  many  hundreds,  how  many  tens,  and  how 
many  units  in  each  of  the  answers. 

The  learning  of  the  addition  combinations  is  a  gradual  process 
accomplished  by  many  repeated  perceptions  on  the  part  of  the  learner. 
Inexperienced  teachers  are  cautioned  not  to  be  discouraged  if  the  same 
pupil  who  has  one  day  given  the  combinations  correctly  misses  them 
at  a  later  date.  This  merely  shows  that  those  paths  in  the  undevel- 
oped little  brain  need  to  be  traversed  again.  Vary  the  work  l>y 
having  pupils  place  and  count  squares,  make  and  count  dots,  or  count 
objects,  real  or  imaginary. 

51 


2 

ADDITION 

NUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

56 

66 

76 

86 

96 

7 

17 

27 

37 

47 

51 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10    20    30    40    50   60   70   80   90   100 

4.  Find  9  on  the  number  table  and  add  2.  Add  2  to 
19.     Add  2  to :  29,  39,  49,  59,  69,  79,  89,  99. 

Show  9  +  2  as  9  +  1,  which  completes  the  ten,  and  1  more  which 
makes  11. 

Call  attention  to  the  relative  position  in  the  number  table  of  9  and 
11, 19  and  21,  etc.,  and  train  pupils  to  associate  with  them  the  thought 
of  2  as  denoting  the  interval  between  them.  Do  this  with  each  com- 
bination, noting  the  respective  intervals. 

5.  19  cents  and  2  cents  =  how  many  cents  ?  How 
many  dimes  and  how  many  cents  over  ? 

6.  Think  of  9  little  birds  and  of  3  more  little  birds 
coming  to  join  them.     Draw  a  picture  of  them. 

7.  49  cents  +  2  cents  equal  hoAV  many  dimes  and  cents? 

8.  99  cents  and  2  cents  =  how  many  cents  ?  How 
many  dimes  and  cents  ?     How  much  more  than  a  dollar  ? 

9.  Add:   93       94      92      95      27      25      96      94      25 

24       24      26      23      91      94      21      24      92 


ADI)1TI(.)N  53 

Tell  how  many  liuiidreds,  liow  many  tens,  and  how 
many  units  in  each  of  the  answers. 

10.  How  many  must  be  added  to  9  to  equal  11  ? 

11.  How  many  must  be  taken  from  11  to  leave  9? 

12.  Show  by  the  number  table  which  is  greater,  9  +  2 
or  2  +  9. 

13.  Find  each  number  in  the  number  table  that  has  9  in 
the  units'  place,  add  8  to  it,  and  remember  the  result. 

Show  that  in  arldiug  3  to  9,  1  completes  the  first  ten  and  the 
remaining  2  make  12. 

14.  Which  is  the  greater,  9  -f  3  or  3  -f-  9  ?     Show  it  on 

the  number  table.    % 

As  each  combination  of  units  is  taken  up,  lead  the  children  to  ob- 
serve that  the  same  result  is  obtained  by  combining  the  numbers  in 
either  order,  and  give  drill  upon  the  combinations  stated  in  each  way. 

15.  Add:  95      96      98      93      92      93      93      34      23 

32      31      21      36      26      35      25      94      91 

How  many  hundreds,  how  many  tens,  and  how  many 
units  in  the  answers  ? 

16.  When  numbers  are  added,  the  result  is  called  their 
Sum.  What  is  the  sum  of  29  and  3?  49  and  3?  89 
and  3? 

17.  Jane  had  19  cents  and  gained  3  cents.  How  many 
cents  had  she  then  ? 

18.  Louise  paid  39  cents  for  a  doll  and  2  cents  for  a 
postage  stamp.     How  much  did  both  cost  ? 

19.  John  had  3  cents  more  than  Thomas.  Thomas  had 
29  cents.      How  much  did  John  liave? 

20.  99  cents  +  3  cents  =  how  many  cents  ?  How  much 
more  than  a  dollar  ? 


^4  ADDITION 

21.  9  pints  +  8  pints  =  liow  many  pints  ?  How  many 
quarts  ? 

22.  How  many  must  be  added  to  9  to  equal  12  ? 

23.  How  many  must  be  added  to  29  to  equal  82  ?     81  ? 

24.  41  is  how  many  more  than  39  ? 

25.  How  many  must  be  taken  from  52  to  leave  49  ? 

26.  9  cents  are  how  many  less  than  11  cents  ?    12  cents  ? 

At  each  lesson  review  combinations  previously  learned.  Lead 
pupils  to  see  that  as  10  +  3  =  13,  9  +  3  must  equal  1  less  than  13, 
9  +  4  must  equal  1  less  than  li,  and  so  on. 

27.  Find  9  and  add  4.     Add  4  to  19.     To  29.     To  69. 

28.  What  is  the  sum  of  89  and  4  ?    69  and  2  ?    39  and  3  ? 

29.  Add:  93  96  98  40  87  i^  95  97  92  97 

42  82  41  99  9L^  92  88  -U  40  22 

30.  Point  out  9  on  tlie  number  table,  and  without 
counting  show  the  number  that  is  4  more  than  9. 

31.  Show  29  and  the  nundjer  that  is  4  more  than  29. 

32.  Show  89  and  the  number  that  is  3  more  than  89. 

33.  Name  the  number  tliat  is  2  more  than  49. 

34.  Name  quickly  the  number  that  is  4  more  than  59. 

35.  Sliow  41  and  the  number  that  is  2  less  than  41. 

36.  Show  53  and  the  number  that  is  4  less  than  53. 

37.  Name  the  number  that  is  4  less  than  13. 

Drill  pupils  on  the  combinations  until  the  mind  furnishes  instanth^ 
the  correct  result.  Then  apply  the  combinations  to  the  facts  of 
childish  experience. 

38.  There  were  29  apples  in  a  basket  and  4  apples 
were  put  in.      How  many  apples  were  in  the  basket  then? 

39.  Lucy  had  19  squares  on  her  desk  and  her  teacher 
gave  lier  3  squares  more.  Hoav  many  squares  had  she 
then? 


ADDITION  55 

40.  William  had  39  cents,  and  Alfred  had  4  more  than 
William.  How  many  cents  did  Alfred  have?  How  many 
dimes  and  how  many  cents  over? 

41.  Add  5  to  each  of  the  numbers  less  than  100  that 
have  9  in  the  units'  place,  and  learn  the  result. 

42.  Add:  98      92      33      94      99      54      47      93      31 

51      43      96      50      50      94      92      54      95 

43.  What  is  the  sum  of  29  and  5  ?    69  and  5  ?    39  and  3  ? 

44.  Show  79  and  the  number  that  is  5  more  than  79. 

45.  Show  39  and  the  number  that  is  5  more  than  39. 

46.  Name  the  number  that  is  5  more  than  49. 

47.  Name  the  number  that  is  4  more  than  89.  Than 
59. 

48.  Mary  may  name  a  number  that  ends  in  9,  and  the 

rest  may  think  of  a  number  that  is   5   more  than  hers. 

What  number  is  3  more  than  hers?     2  more?     4  more? 

Make  a  general  exercise  of  the  work  of  Ex.  48.  As  new  numbers 
are  taken  up  use  similar  exercises. 

49.  Think  of  74  and  give  the  number  that  is  5  less 
than  74. 

50.    may  give   a    number   ending   in  4,   and    the 

rest  may  give  the  number  tliat  is  5  less. 

51.  There  were  29  people  in  a  car,  and  5  got  on  the  car 
at  the  station.     How  many  persons  were  in  the  car  then? 

52.  There  was  a  school  of  39  children,  and  4  new  pupils 
were  brought  into  it.  How  many  pupils  were  in  the 
school  then  ? 

53.  Add  6  to  9,  and  to  each  of  the  numbers  less  than 
100  that  end  in  9. 

54.  Add:     96     94     45     97     94     98     54     61     44     97 

53     Qd     91     62     35     61     93     95     92     50 


56  ADDITION 

55.  Show  69  and  the  number  that  is  6  more  than  69. 

56.  Show  49  and  the  number  that  is  6  more  than  49  ;  4 
more  than  49 ;  2  more  than  49 ;  5  more  than  49 ;  3  more 
than  49. 

57.  Think  of  79  and  tell  the  number  that  is  6  more 
than  79  ;   3  more  ;   5  more  ;   2  more  ;  4  more. 

See  Ex.  48  and  the  note  following. 

58.  What  number  is  6  more  than  59?  2  more?  5 
more?     3  more? 

59.  What  number  is  6  less  than  75?     6  less  than  95? 

60.  How  many  must  be  added  to  49  to  equal  55? 

61.  Six  pupils  of  a  school  stayed  at  noon.  The  39 
other  pupils  went  home  to  dinner.  How  manj  pupils 
were  there  in  all? 

62.  What  is  the  difference  between  9  and  14  ?  Between 
19  and  25  ?  29  and  33  ?  69  +  ?  =  74  ?  69  +  ?  =  72  ? 
69  +  ?  =  75  ? 

63.  33  cents  are  how  many  more  than  29  cents  ? 

64.  Arthur  found  19  eggs  in  one  nest  and  6  eggs  in 
another.     How  many  did  he  find  in  all  ? 

65.  Lizzie  has  39  cents  and  needs  6  cents  more  to  buy 
the  doll  she  wants.     What  is  the  price  of  the  doll  ? 

66.  Draw  a  line  on  the  board  3  feet  long.  It  is  a  yard 
long.      We  can  call  it  either  1  yard  or  3  feet. 

Let  the  boys  draw  lines  of  varions  lengths  on  the  floor,  marking  off 
the  foot  units  and  the  yard  units.  Use  them  in  working  out  the  fol- 
lowing,  and  refer  to  them  whenever  the  children's  imaginations  fail 
to  give  the  correct  ideas  of  feet  and  yards. 

67.  How  many  feet  in  3  yards  ?  4  yards  ?  2  yards  ? 
6  yards  ?     5  yards  ?     8  yards  ? 


ADDITION  57 

68.  I  low   many   yiirds   in   O   feet?      12   feet?     9   feet? 
15  feet  ?     18  feet  ?  ^  21  feet  ?     27  feet  ?     30  feet  ? 

69.  t)   feet  +  o   feet  =  how  many   feet  ?      How   many 
yards  ? 

70.  9  feet  +  3   feet  =  how  many   feet  ?      How   many 
yards  ? 

71.  How  many  yards  in  12  feet  +  3  feet  ?      15  feet  +  3 
feet? 

72.  How  many  feet  in  1  yard  +  1  foot  ?     2  yards  +  2 
feet  ? 

73.  How  many  feet  in  4  yards  +  1  foot  ?     3  yards  -f-  2 
feet  ? 

74.  2  yards  +  3  feet  =  how  many  yards  ? 

75.  7  feet  =  how  many  yards  and  how  many  feet  ov^er  ? 

76.  How  many  yards  and  feet  in  11  feet?     13  feet? 
8  feet  ? 

77.  9  feet  -f  1  feet  =  liow  many  yards  and  feet  ? 

78.  How  mucli  do  9  feet  +  5  feet  hick  of  benig  5  yards  ? 

79.  (9  +  5)-2  =  ?     (19h-5)-2  =  ?     (9  +  3)-r-2  =  ? 

80.  To  each  of  the  numbers  less  than  100  that  have  9 
in  the  units'  phice  add  7. 

81.  What  is  the  sum  of  19  and  7  ?    39  and  7?    89  and  7  ? 

82.  Add:     93       98       91       45       9G       97       93       91 

7461729352627438 

Tell  how  many  hundreds,  how  many  tens,  and  how  many 
units  in  each  answer. 

83.  Write  91  and  73  and  find  their  sum.     98  +  40  =  ? 

84.  Show  9  and  the  number  that  is  7  more  than  9. 

85.  Show  89  and  the  number  that  is  7  more  than  89. 


58  ADDITION 

86.  Name  the  number  that  is  7  more  than  49.  5  more  ; 
3  more  ;   6  more  ;   2  more  ;   4  more. 

87.  What  number  is  7  less  than  16  ?  7  less  than  46  ? 
7  less  than  86  ?     5  less  than  76  ?     4  less  than  63  ? 

88.  What  number  must  be  added  to  39  to  equal  46  ? 

89.  There  are  9  persons  at  home  in  Mr.  Smith's  family. 
When  they  have  7  visitors,  how  many  persons  are  in  the 
house  ? 

90.  9  pints  +  7  pints  =  how  many  quarts  ? 

91.  Thomas   had   19   cents   and  his  father  gave  him 

7  cents.     How  many  cents  had  he  then  ? 

92.  49  men  were  working  on  a  building  when  7  other 
men  Avere  hired  to  help.  How  many  workmen  were  there 
in  all  ? 

93.  Find  each  of  the  numbers  that  have  9  in  the  units' 
place  and  are  less  than  100,  and  add  8  to  it. 

94.  9  squares  and  8  squares  =  how  many  squares  ? 

95.  Add:  93       95       87       86       92       95       24       96 

8684919374439181 

96.  Show  9  and  the  number  that  is  8  more  than  9. 
6  more  than  9.     4  more  than  9.     7  more  than  9. 

97.  Think  of  29.  What  number  is  8  more  ?  5  more  ? 
2  more  ?    7  more  ?    4  more  ? 

98.    i^i^y  Ht'ime  ^  number  ending  in  9,  and  otliers 

may  add  to  it  numbers  less  than  9. 

99.  19  books  were  on  a  shelf  and  8  others  were  added. 
How  many  books  were  there  then  ? 

100.  89   persons  were    in  a   meeting.     When  8   other 
persons  came  in,  how  many  were  present  ? 

101.  One  coAV  gave  9  quarts  of  milk  and  another  gave 

8  quarts.      How  much  milk  did  l)oth  cows  give  ? 


ADDITION 


59 


102.  9  ft.  -f-  8  ft.  =  liow  many  yd.  and  ft.  ? 

Contractions  of  denominations  will  be  used  hereafter  interchange- 
ably with  the  complete  words. 

103.  ]\lary  picked   9   pt.   of  berries  and  Anna  picked 
7  pt.     How  many  pt.  did  both  pick  ?     How  many  qt.  '! 

104.  Add  9  to  9,  and  to  each  of  the  nnmbers  less  than 
100  that  end  in  9. 

105.  Add:     95       98       9^       97       93      49       93      92 

93       91       35       92       94       90       64       87 


106.  Show  on  the  number  table  29  and  the  number 
that  is  9  more  than  29. 

107.  Show  the  number  that  is  9  more  than  49.  9  more 
than  79.  9  more  than  39.  8  more  than  59.  7  more  than 
69.     6  more  than  79.     5  more  than  89.     4  more  than  29. 

108.  29  cents  4-  9  cents  =  how  many  cents  ? 

109.  99  cents  +  9  cents  =  how  much  more  than  a  dollar? 

110.  Anna  and  Mary  each  picked  9  pt.  of  berries. 
How  many  qt.  did  both  pick  ? 

111.  How  many,  sums  did  you  find  in  working  Ex.  95  ? 
How  many  in  working  Ex.  105  ?     How  many  in  both  ? 

BLACKBOARD    EXERCISE. 


Practice  rapid  adding  of  each  figure 
on  the  edge  of  the  square  to  the  one 
in  the  middle.  Then  change  9  to  19, 
29,  39,  etc. 


5 

2 

7 

8 

9 

3 

6 

4 

9 

50  ADDITION 

112.  John  has  9  marl)les  and  James  has  7  marbles. 
How  many  have  both  ? 

113.  Make  story  prol)lems  in  addition  with  9  as  one  of 
yonr  nnmbers. 

114.  Add:  72  96  93  92  45  93  68  92  95  64 

35326584947291213291 

115.  How  many  hundreds,  how  many  tens,  and  how 
many  units  in  the  number  287?  In  307?  422?  330? 
976  ? 

116.  Write  a  number  that  has  3  hundreds,  8  tens,  and 
7  units. 

117.  Write  a  number  made  of  2  hundreds,  6  units,  and 
3  tens. 

118.  Write  a  number  having  4  in  the  units'  phice,  6  in 
the  tens'  place,  and  7  in  the  hundreds'  place. 

119.  Write  a  number  having  9  in  the  tens'  place,  3  in 
the  hundreds'  place,  and  4  in  the  units'  place. 

It  should  be  explained  that  of  the  16  obtained 

120.  To  29      by  adding  9  and  7,  the  6  units  should  be  written 
add   7      in  the  units'  place,  and  the  ten  should  be  combined 

with  the  2  tens.  The  child  should  be  led  to  see 
that  the  36  obtained  in  this  way  is  the  same  result  as  that  which  he 
got  at  first  by  counting. 

121.  Add:  49      89      69      69      39      49      29      59      69 

J7jr_7272727272716 

69    27    326    235    144    63    349    129 
27    49    469    749    629    29    542    315 

While  the  class  are  taking  the  form  work  and  fractions  which  fol- 
low, keep  the  addition  work  clear  in  their  minds  by  short  drills  and 
by  giving  examples  like  the  preceding  for  seat  work. 

122.  Fold  an  inch  square  of  pa2)er  into  4  equal  parts. 


ADDITION 


61 


123.  When  anything  is  divided  into  4  equal  parts,  what 
is  each  part  called  ? 

124.  When  anything  is  divided  into  5  equal  parts,  what 
is  each  part  called  ?     Ans.  |. 

125.  Place  triangles  as  in  Fig.  1. 
How  many  triangles  does  it  take  to 
copy  Fig.  1  ?  •  Fig.  1 

126.  Take  away  -J  of  your  figure.  How  many  fifths  are 
left  ?  Put  back  the  ^  and  take  away  |.  How  many  fifths 
are  left  ? 

127.  Build  Fig.  2  with  triangles 
Show  i  of  Fig.   2.     -I  of  it.     I  of  it.  -p^^  2 

128.  How  many  triangles  did  you  use  in  building  Fig, 
2  ?     How  many  square  inches  would  they  equal  ? 

129.  Build  Fig.  3  with  triangles.  Which  is 
larger,  Fig.  2  or  Fig.  3  ? 

130.  How  many  triangles  did  you  use  in  mak- 
ing Fig.  3  ?  How  many  square  inches  would 
they  equal  ?  Fig.  3. 

131.  Can  you  take  away  one  triangle  from  Fig.  3  so  as 
to  leave  a  large  square  ? 

132.  Show  ^  of  Fig.  3.  Show  f  of  it  and  tell  how 
many  fifths  are  left.     Show  |  and  what  is  left. 

133.  Put  5  triangles  into  a  figure  different  from  any  in 
the  book.     Show  |-  of  it ;  |-  of  it. 

Let  the  pupils  carry  on  the  processes  of  building  figures  and  part- 
ing, wholiiig,  and  parting  them  again  as  long  as  the  exercise  is  inter- 
esting and  instructive.  If  the  quicker  ones  anticipate  the  "work  to 
come  and  show  sixths,  seyentbs.  and  eighths,  so  much  the  bettei*.    The 


62 


ADDITION 


presentations  of  the  teacher  and  of  the  book  must  be  in  an  orderly 
progress-ion,  but  children  should  be  encouraged  to  make  their  own 
discovei'ies  freely. 

134.    Draw    a    vertical    line    5    inches    lono*. 


Mark  it  off  into  inches.     Show  ^  of  it.    -|. 


5* 


2 

5' 


1  incli 


135.  Draw  a  horizontal  line  10  inches  long 
and  show  -J  of  it.  Show  |  of  it.  Show  |-  of  it. 
ShoAv  I  of  it. 

136.  HoAV  many  inches  long  is  a  line  that  is 
|-  of  10  inches  long  ?  How  long  is  -|  of  a  10- 
inch  line  ?  I  of  a  10-inch  line  ?  -|  of  a  10-inch 
line  ? 

137.  10   pupils  may  stand  in   a  line. 


1 

5 


of 


them  at  this  end  of  the  line  may  step  forward. 
2  inches    Take  places  again.      |  at  the  other  end  of  the 
line  may  step  forward.     |   ul  10  =  how  many  ? 
I  of  10  =  lit)W  many  ? 

138.    Turn  to  the  number  table  and  show  ^  of 


50.      I  of  50. 

5 


f  of  50.     I  of  50.     i  of  100. 


3  inches 


Fig.  4 


4  inches 


139.  Place  triangles  so  as  to  make  Fig.  4. 
How  many  square  inches  ' 
does  it  equal?  Hoay 
many  triangles  in  Fig. 
4  ?  Separate  it  into 
halves.  How  many  triangles  in  each  half? 
Put  the  halves  together  again. 

140.  When  anything  is  divided  into  6  equal 
parts,  what  is  each  part  called  ? 

141.  Show  i  of  Fio-.  4.     I  of  it.     Show  |  of 


6 
1  9 


it.     Whicli  is  greater,  1  or  |  ?    ^  or  #  r    A  or 


2  9     1 
6   •      2 


3  '/» 

G   • 


5  inches 


1    _L  2 
G     1^  6 


142 

many  Oths  ? 


how  many  Oths? 


1  +  I  =  lu.w 


ADDITION 


63 


Fig.  5 


Fig.  6 


Fig.  7 


143.  Copy  Figs.  5,  6,  and  7  by  placing  triangles. 

144.  Can  you  take  away  two  triangles  from  Fig.  5  so 
as  to  leave  2  square  inches  ? 

145.  Can  you  take  away  two  triangles  from  Fig.  6  so 
as  to  leave  2  square  inches  ?  Notice  the  triangles  whose 
square  corners  are  at  the  center  of  the  hgure. 

146.  Can  you  take  away  two  triangles  from  Fig.  7  so 
as  to  leave  a  large  square  ? 

147.  Can  you  take  away  two  triangles  from  Fig.  7  so 
as  to  leave  2  square  inches  ? 

148.  From  Fig.  7  take  away  J,  and  show  how  many 
sixths  are  left. 

149.  How  many  sixths  are  left  if  you  take  away  |  ?   |  ? 

150.  I  —  -1  =  how  many  sixths  ?  |  —  |  —  how  many  sixths  ? 
f  —  f  =  how  many  sixths  ?  |  —  f  =  how  many  sixths  ? 

151.  Place  triangles  as  in  Fig-.  8. 
Can  you  take  away  four  triangles 
and  leave  1  square  inch  ? 

152.  Separate  Fig.  8  into  halves 
by  taking  away  the  lower  row  of 
triangles.  How  many  sixths  in 
each  half  ? 

153.  Place  trianafles  as  in  Fig-.  9. 
Separate  the  figure  into  three  equal 
figures  of  the  same  shape.  How 
many  sixths  in  each  third?  Fis.  9 


Fig.  8 


64 


ADDITION 


Fig.  10 


154.  Place  triangles  as  in  Fig.  10. 
Separate  the  figure  into  halves.    ^  =  i'- 

155.  Place  six  triangles  so  as  to  make 
a  different  figure  from  any  in  the  book, 
and  divide  it  into  halves. 

156.  Make  a  figure  of  six  squares.  Show  1  of  it.  Show 
I  of  it.      Show  I  or  ^  of  it.     Show  |  of  it. 

157.  Draw  a  vertical  line  6  inches  long.  Show  1  of  it. 
Show  I  of  it.     Sliow  I  of  it.     What  does  |  of  it  equal? 

158.  Figure  11  is  a  rectangle.     How  many  sides  has  it? 

How  many  corners  ?     What  kind  of  corners 

has  it  ? 

Let   pupils   find    rectangular   surfaces,  as   window 
panes,  blackboards,  etc. 

159.  Draw  a  rectangle  (3  inches  long  and  1  inch  wide, 
and  divide  it  into  6ths  by  vertical  lines. 

160.  Draw  a  rectangle  whose  horizontal  lines  are  each 
5  inches  long  and  vertical  lines  1  inch  long,  and  divide  it 
into  fifths. 

161.  Draw  a  rectangle  3  inches  long  and  2  inches  wide, 
and  divide  it  into  6ths. 

162.  Draw  a  picture  of  a  pie  cut  into  sixths. 

163.  Turn  to  the  number  table  and  show  1  of  60.  |  of 
60.     I  of  60.     I  of  60. 

164.  Add:  226     433     659 

319     529     124 


Fig.  11 


649     377 
32     519 


739 
249 


429 
535 


349 

6:!  8 


165.  Add  :  437       How  many  hundreds,  liow  many  tens, 

641  and  how  many  units  in  the  answer  ? 

166.  10  hundreds  make  1  thousand.     Which  place  do 
the  thousands  have  ? 


ADDITION  65 

167.  How  many  thousands,  how  many  hundreds,  how 
many  tens,  and  how  many  units  in  the  number  7654  ? 
4326?     6304?     3829?     5340?     2002? 

168.  Write  a  number  made  of  5  units,  2  hundreds,  7 
thousands,  and  8  tens. 

169.  Write  and  read  a  number. which  has  5  in  the 
fourth  place,  8  in  the  third  place,  3  in  the  second  place, 
and  1  in  the  first  place. 

170.  Write  a  number  of  4  places  and  tell  what  is  in 
each  place. 

171.  Write  4351  under  5437  and  add  them.  Why  is  it 
best  to  put  units  under  units,  tens  under  tens,  hundreds 
under  hundreds,  and  thousands  under  thousands  ? 

172.  Add:    3219      8647      7935      6396      3456      1639 

1234      1239      1249      1272      1939      1924 

173.  Add  3  to  8,  and  add  3  to  each  of  the  numbers  less 
than  100  that  end  in  8. 

Show  that  in  adding  3  to  8,  2  completes  the  first  ten,  and  the  re- 
maining 1  makes  11. 

174.  Find  the  sum  of  18  and  3.     48  and  3.     78  and  3. 

175.  Show  on  the  number  table  the  number  that  is  3 
more  than  8.  3  more  than  28.  3  more  than  88.  3  more 
than  58.     3  more  than  38. 

176.  8  ft.  +  3  ft.  =  how  many  yd.  and  ft.? 

177.  18  ft.  +  3  ft.  =  how  many  yd.? 

178.  There  were  38  sheep  in  a  flock  and  3  sheep  were 
added  to  it.     How  many  were  there  then  ? 

179.  John  worked  28  examples  on  Monday,  and  on 
Tuesday  he  worked  3  more  than  on  Monday.  How  many 
did  he  work  on  Tuesday  ? 

HORN.    ARITH.  5 


66 


ADDITION 

« 

180. 

Add: 

325 

1928  2878 

2463 

349 

284 

813 

819 

633   913 

928 

815 

631 

358 

181.  Find  the  sum  of  1328  and  2843.  4928  and  2953. 

182.  Find  the  sum  of  80  and  30.  90  and  30.  90  and  50. 

183.  Find  the  sum  of  8  and  4.  28  and  4.  38  and  4. 
68  and  4.  88  and  4.  18  and  4.  58  and  4. 

See  suggestion  after  Ex.  173. 

184.  Show  on  the  number  table  the  number  that  is  4 
more  than  78.     4  more  than  48.     4  more  than  58. 

185.  Add:     827     869     1238     1835     1598     1318     823 

)   151    M^     1^     1429     4123     4824     448 

186.  Mary  found  38  peaches  under  one  tree  and  4 
peaches  under  another.  How  many  peaches  did  she  find 
in  all  ? 

187.  James  went  to  the  grocery  to  buy  sugar.  In 
bringing  it  home  he  spilled  4  pounds.  He  brought  home 
8  pounds.     How  much  did  he  buy  ? 

188.  There  were  28  pounds  of  butter  in  a  jar  and  4 
pounds  more  were  put  into  the  jar.  How  many  were  in 
it  then  ? 

189.  Find  the  sum  of  80  and  40.    90  and  40.    90  and  70. 

190.  8  ft.  4-  4  ft.  =  how  many  yd.? 

191.  8  pt.  +  4  pt.  =  how  many  qt.? 

192.  How  many  quarts  make  a  gallon  ? 

Let  the  children  use  quart  and  gallon  measures  and  find  out  the 
fact  for  themselves. 

193.  How  many  quarts  make  2  gallons  ?  3  gallons  ? 
4  gallons  ?     5  gallons  ? 

194.  How  many  quarts  in  2  gallons  and  1  quart  ?  3  gal. 
and  2  qt.? 


ADDITION  67 

195.  There  were  2  gal.  and  3  qt.  of  molasses  in  a  jug, 
and  Mary  used  a  quart  of  it  to  make  candy.  How  many 
qt.  were  left  ? 

196.  How  many  gal.  in  8  qt.?    12  qt.?    20  qt.?    16  qt.? 

197.  How  many  gal.  and  how  many  qt.  over  in  9  qt.? 
11  qt.?     13  qt.?     15  qt.? 

198.  How  many  gal.  in  8  qt.  +  4  qt.?     9  qt.  -|-  7  qt.? 

199.  9  qt.  less  1  qt.  =  how  many  gal.? 

200.  Add  5  to  each  of  the  numbers  less  than  100  ^vliose 
unit  figure  is  8. 

201.  Find  the  sums  : 

873   729   8386   628   989   8359   648   933   445 
524   943   5353   843   250   1680   289   488   884 

202.  8  ft.  H-  5  ft.  =  how  many  yd.  and  ft.  ? 

203.  58  lemons  were  in  a  box.  If  5  more  lemons  were 
placed  in  the  box,  how  mau}^  would  it  contain? 

204.  Mr.  Smith  set  out  5  new  trees  in  his  orchard, 
which  already  had  88  trees.  How  many  trees  were  there 
then? 

205.  A  book  cost  28  cents  and  a  tablet  cost  5  cents. 
How  much  did  they  both  cost? 

206.  Find  the  sum  of  80  and  50.     80  +  30.     90  +  90. 

207.  8  qt.  +  5  qt.  =  liow  many  gal.  and  qt.  ? 

208.  2  gal.  +  4  qt.  =  how  many  qt.  ?     How  many  gal.  ? 

209.  If  you  drank  a  pint  of  milk  every  day,  how  many 
pints  would  you  drink  in  2  weeks?     How  many  quarts? 

210.  Add  6  to  some  numbers  that  end  in  8,  and  tell 
how  many  tens  and  how  many  units  in  each  answer. 

211.  If  you  picked  8  apples  from  a  tree  and  should 
pick  off  6  more,  how  many  would  you  have? 


68  ADDITION 

212.  When  John  worked  18  problems  in  the  morning 
and  6  in  the  afternoon,  how  many  did  he  work  in  the 
whole  day? 

r. 

213.  8  ft.  +  6  ft.  =  how  much  less  than  5  yd.  ? 

214.  Find  the  sums  : 

846        3858        4888         3884         4898         8888         3618 
698        5596        4456         5659         9586         3456         4845 

How  many  thousands,  how  many  hundreds,  how  many 
tens,  and  how  many  units  in  each  answer? 

215.  Write  and  read  a  number  which  means  1  thou- 
sand, 3  hundreds,  5  tens,  7  units. 

216.  Write  a  number  that  means  2  thousands,  5  hun- 
dreds, 0  tens,  4  units. 

217.  Write  a  number  that  is  made  of  11  thousands,  5 
hundreds,  3  tens,  0  units. 

218.  Put  4  tens,  5  hundreds,  3  units  into  one  number. 

219.  Put  6  units,  7  hundreds,  8  tens  into  one  number. 

220.  Put  2  thousands,  1  unit,  6  tens,  4  hundreds  into 
one  number. 

221.  Put  into  one  number  12  thousands,  2  units,  4 
hundreds,  3  tens. 

222.  Put  into  one  number  25  thousands,  4  tens,  1  hun- 
dred, 6  units. 

223.  8  qt.  +  6  qt.  =  how  many  gal.  and  qt.  over? 

224.  Add  7  to  several  numbers  whose  unit  figure  is  8. 

225.  Harriet  is  8  years  old  and  Lucy  is  7  years  older. 
How  old  is  Lucy? 

226.  A  carriage  cost  88  dollars,  and  a  horse  cost  7  dol- 
lars more  than  the  carriage.  How  nuich  did  the  horse 
cost  ? 


ADDITION  69 

227.  8  ft.  and  7  ft.  =  how  many  yd.  ? 

228.  Make  prublenis  using  7  and  numbers  that  end  in  8. 

229.  Add :  5898      3968      2897      5848      8989      25868 

9697      1497      3958      3697      4069      48036 

230.  What  is  the  sum  of  80  and  70?  80  and  60?  80 
and  30  ?     80  and  40  ? 

231.  8  qt.  +  7  qt.  =  how  many  qt.  ?  How  much  less 
than  4  gah? 

232.  Add  8  to  some  numbers  whose  unit  iigure  is  8. 

233.  Anna  has  2  dolls  each  of  which  cost  8  cents. 
How  much  did  they  both  cost  ? 

234.  Mr.  Smith  has  2  horses  each  of  which  cost  80 
dollars.     How  much  did  they  both  cost  ? 

235.  8  qt.  +  8  qt.  =  how  many  gal.? 

236.  8  ft.  +  8  ft.  =  how  many  yd.  and  ft.? 

237.  Find  the  sums  : 

867      2183      8482      8623    "  8623      6683      3468      9817 
832      9434      8351      8351      7986      8275      8522      6528 

238.  John  drew  a  line  5  ft.  long.  How  much  over  a 
yard  was  it  in  length  ? 

239.  Samuel  drew  a  line  8  ft.  long  and  William  drew 
one  3  yd.  long.  How  much  longer  was  William's  line 
than  Samuel's  ? 

240.  Draw  on  the  board  a  square  1  ft.  long  and  1  ft. 
wide.  If  you  had  a  string  long  enough  to  lie  all  around 
on  the  line  that  bounds  the  square,  how  many  ft.  long 
would  the  string  be  ?  How  many  yd.  and  how  many  ft. 
over  ? 


70 


ADDITION 


241.  A  rectangle  is  2  ft.  long  and  1  ft.  wide.  Draw  a 
small  picture  of  the  rectangle,  and  find  how  many  ft. 
around  it.     How  many  yards  ? 

See  that  the  proportions  of  the  "  picture  "  are  correct. 

242.  Draw  a  picture  of  a  rectangle  3  ft.  long  and  1  ft. 
wide,  and  show  how  many  ft.  in  the  distance  around  it. 

243.  Draw  a  picture  of  a  rectangle  whose  horizontal 
lines  are  3  ft.  long  and  whose  vertical  lines  are  2  ft.  long, 
and  tell  how  far  it  is  around  it. 

244.  Add  9  to  each  number  less  than  101  whose  unit 
figure  is  8. 

245.  Show  in  what  direction  from  28  in  the  number 
table  is  the  number  that  is  9  more  than  28  ;  9  more  than 
38  ;  9  more  than  48. 

246.  How  much  will  a  span  of  horses  cost  if  one  horse 
costs  80  dollars  and  the  other  90  dollars  ? 

247.  8  qt.  4-  9  qt.  =  how  many  qt.  more  than  4  gal.? 

248.  28  cents  +  9  cents  =  how  many  dimes  and  cents  ? 

249.  8  ft.  +  9  ft.  =  how  many  ft.  less  than  6  yd.? 

250.  Add:  2478      2348      8124       9825       8483       1628 

8619      3798      7318       8938       9329         849 


BLACKBOARD    EXERCISE 


6 

9 

4 

3 

8 

7 

5 

8 

2 

Let  each  of  the  numbers  on  the 
edge  of  the  square  be  added  to  the 
number  in  the  middle.  Then  use  18, 
28,  etc.,  instead  of  8.  If  the  children 
cannot  give  any  combination  at  sight, 
review  it. 


ADDITION 


71 


251.  Find  the  largest  even  number  that  is  less  than  29 
and  add  7  to  it. 

252.  Add  8  to  the  largest  even  number  that  is  less 
than  49. 

253.  Five  boys  have  how  many  toes  ?  Nine  boys  have 
how  many  toes? 

254.  Give  all  the  multiples  of  10  tliat  are  less  than  100. 

255.  Find  the  number  that  is  1  less  than  the  third 
multiple  of  10.  Add  6  to  it.  Add  4  to  it.  Add  7  to 
it.     Add  9  to  it.     Add  5  to  it. 

256.  Find  the  number  that  is  2  less  than  the  fifth  mul- 
ti[)le  of  10  and  add  9  to  it.  Add  3  to  it.  Add  7  to  it. 
Add  4  to  it.     Add  8  to  it. 

257.  10  twos  =  ?  Find  tlie  number  that  is  1  less  than 
10  twos  and  add  4  to  it.  Add  7  to  it.  Add  5  to  it. 
Add  8  to  it. 

258.  Draw  a  rectangle  5  in.  long  and  4  in.  wide,  and 
tell  how  far  it  is  around  it. 

259.  The  line  around  a  figure  is  called  its  Perimeter. 
Draw  a  rectangle  4  in.  long  and  3  in.  wide,  and  tell  how 
long  its  perimeter  is. 

260.  How  long  is  the  j)erimeter  of  a  rectangle  which  is 
3  in.  lono-  and  3  in.  wide  ?     Draw. 

Recall  fractious  previously  learned. 

261.  Copy  Fig.  12  by  placing  triangles. 
How  many  triangles  in  Fig.  12  ? 

262.  Show  -^  of  the  figure.     Show  f  of 
the  figure.     Show  ^  of  it.      How  many 


■^      2.  _^  |.  _  \lQ^^Y  many  sevenths  ? 

9 


remain  r 

|-  +  I  =  how  many  sevenths  ? 


Fig.  12 


72  ADDITION 

263.  Copy  Fig.  13.  How  many  tri- 
angles in  the  figure  ? 

264.  Take  away  ^  of  the  figure  and 
show  how  many  sevenths  remain.  Take 
away  |-  and  show  how  many  seventlis 
remain.     Take  away  ^. 

_?        6_4_1       4_2_1       2_2_?_ 

7"       7         T  ~"  7        1        Y  ~  7        7         7   ""  7 

266.  Place  7  triangles  so  as  to  make  a  figure  different 
from  those  in  the  book,  and  show  sevenths  of  it. 

267.  Draw  a  rectangle  7  in.  long  and  1  in.  wide,  and 
find  the  length  of  its  perimeter. 

268.  7  +  4  =  ?     Add  4  to  each  of  the  numbers  in  the 
number  table  whose  unit  figure  is  7. 

269.  17  cents  +  4  cents  =  how  many  dimes  and  cents  ? 

270.  70    cents  +  40    cents  =  how   many    cents  ?       How 
many  dollars  and  dimes  ? 

271.  7  qt.  +  4qt.  =  how  many  gal.  and  qt.  ? 

272.  John  has  47  dollars  and  needs  4  dollars  more  to 
buy  a  bicycle.     What  is  the  price  of  the  bicycle  ? 

273.  Add :  7425      7378      17849      8795      2135      8123 

4539      1839      29847      9249      1719      6427 

274.  Add  5  to  each  of  the  numbers  in  the  number  table 
whose  unit  figure  is  7. 

275.  7  cents  +  5  cents  =  how  many  dimes  and  cents  ? 

276.  70  cents  +  50  cents  =  how  many  dollars  and  cents  ? 

277.  7  pt.  -f  5  pt.  =  how  many  qt.  ?      7  ft.  +  5  ft.  = 
how  many  yd.? 

278.  A  line  17  in.   long  was  lengthened  5  in.       How 
many  ft.  and  in.  long  was  it  then  ? 


ADDITION  73 

279.  5  qt.  of  milk  were  poured  into  a  can  that  already 
held  27  qt.  of  milk.  How  many  gal.  were  in  the  can 
then  ? 

280.  Add :  7642      1729      8757       9562       8967       3724 

5251      8449      5435       8737       7724       8573 

281.  Add  6  to  each  number  smaller  than  108  whose 
unit  figure  is  7. 

282.  60  +  70  =  ?     90  +  40  =  ?     80  +  e50  =  ?     80  +  70  =  ? 

283.  Find  the  sum  of  147  and  6.     327  and  6.     437  and  6. 

284.  John  had  107  cents  and  gained  6  cents.  How 
many  dollars,  dimes,  and  cents  had  he  then  ? 

285.  How  many  dollars,  dimes,  and  cents  in  the  sum  of 
207  cents  +  6  cents  ? 

286.  407  cents  +  6  cents  =  ?     217  cents  +  6  cents  =  ? 

287.  427  cents  +  6  cents  =  ?     967  cents  +  6  cents  =  ? 

288.  Add:  3754       325       9626       8768       9387        272 

8927       947         947       1207       8904      9986 

289.  Add  7  to  each  number  smaller  than  108  whose 
unit  figure  is  7. 

290.  70  +  70  =  ?     80  +  80  =  ?     90  +  90  =  ? 

291.  Find  the  sum  of  187  and  7.    157  and  7.    277  and  7. 

How  many  dollars,  dimes,  and  cents  in  the  sum  of  : 

292.  314  cents  +  7  cents  ?       295.    227  cents  +  4  cents  ? 

293.  287  cents  +  7  cents  ?       296.    357  cents  +  6  cents  ? 

294.  537  cents  +  7  cents  ?       297.    947  cents  +  7  cents  ? 

298.  Name  the  days  in  the  week.  How  many  days  in 
2  weeks  ?  3  weeks  ? 


74  ADDITION 

299.  Mary  made  a  visit  of  17  days.  Then  her  mother 
allowed  her  to  stay  a  week  longer.  How  many  days  in 
all  did  she  stay  ? 

300.  27  days  +  1  week  =  hoAV  many  days  ? 

301.  A  line  7  ft.  long  was  lengthened  7  ft.  ITow  nnich 
did  it  lack  then  of  being  5  yd.  long  ? 

302.  How  wide  is  a  square  that  is  7  in.  long?  Draw 
a  square  7  in.  long  and  hnd  the  length  of  its  perimeter- 

303.  Mr.  Smith  is  27  years  old,  and  j\lr.  Brown  is 
7  years  older.     How  old  is  Mr.  Brown  ? 

304.  Add:  2736      7826      4727      7268      50627      9773 

9729      7273      9737      7981        7197      6576 

305.  Add  8  to  each  number  less  than  118  ^vhose  unit 
ligure  is  7. 

306.  70  +  80  =  ?  70  +  50  =  ?  70  +  60  =  ? 

307.  Find  tlie  sum  of  137  and  8,       267  +  8.       967  +  8. 

308.  7  ft.  +  8  ft.  =  how  many  yd.  ? 

How  many  dollars,  dimes,  and  cents  in  the  sum  of  : 

309.  547  cents  +  8  cents  ?        312.    187  cents  +  4  cents  ? 

310.  627  cents  +  8  cents  ?       313.    189  cents  +  7  cents  ? 

311.  917  cents  +  8  cents  ?       314.    218  cents  +  7  cents  ? 

315.  It  is  the  custom  to  write  364  cents  *^3.64  and  to 
call  it  3  dollars  and  64  cents.     Read:  84.84  ;  $9.87. 

316.  Add:. $7.47     $8.27      $25.37      $48.47      $18.57 

2.38        1.48         97.28         23.27         25.36 

317.  The  point  Avhich  separates  dollars  and  cents  is 
called  a  Decimal  Point.  In  writing  columns  of  dollars 
and  cents  to  be  added  or  subtracted,  why  is  it  best  to  put 
the  points  in  a  vertical  line  ? 

318.  Add  9  to  each  number  that  is  less  than  118  and 
lias  7  in  the  units'  place. 


ADDITION 


75 


319.  What  is  the  sum  of  12T  and  9  ?    23T  +  0  ?    487  +  9  ? 

320.  AVhat  is  the  sum  of  TO  and  90  ?     TO  and  40  ? 

321.  '^  1.2T  +  9  cents  =  ?     -^  2.3T  +  9  cents  =  ? 

322.  T  (|l.  +  9  qt.  =  how  many  gal.? 

323.  T  ft.  +  9  ft.  =  how  many  yd.  and  ft.? 

324.  T  pt.  -f  9  pt.  =  how  many  qt.?      How  many  gal.? 

325.  Add:  .it54.5T     -^BG.BT     ^lo.TT     i36.5T      !i?18.9T 

9.29        19.3T        98.2T        23.3T        18.99 


Practice  sight  addition  of  tlie  num- 
bers on  the  edge  of  the  square  to  the 
number  in  the  middle,  and  then  sub- 
stitute for  7  other  numbers  whose  unit 
figure  is  7. 


326.  What  time  is  it  when  the  hour  hand  of  the  clock 
is  on  Y  ? 

327.  V  stands  for  5  in  Roman  notation.      On  what  page 
does  the  5th  chapter  of  this  book  begin? 

Show  that  two  Vs  placed  as  follows  ^  make  X  or  10. 

328.  I  written  after  V  means  I  added  to  V,  or  6.     What 

does   YII   mean?     VIII?     Find  the   heading  of    the  6th 

chapter  in  this  book.      The  8th.      The  Tth. 

Explain  that  when  the  smaller  numeral  is  written  after  the  greater 
their  sum  is  to  be  found. 

329.  Read   XX.       XVI.       XVII.       XVIII.      XXV. 
XXVI.    XXVII.    XXVIII.    XXX.    XXXV.   XXXVII. 

330.  AVrite  in  Roman  notation  15,  IT,  23,  25,  28,  30. 


7(3    ■  ADDITION 

331.  Write  in  Roman  notation  the  first  even  number 
in  the  second  ten. 

332.  Write  in  Roman  notation  the  nnmber  that    is   7 
more  than  the  3d  multiple  of  10. 

333.  Add  5  to  several  numbers  whose  unit  figure  is  6. 

334.  Find  the  sum   of    186  +  5.       210  +  5.      296  +  5. 

335.  Add:  111.26  -i^7.36  $59.36  -^18.76  134.55  182.75 

13.45      5.45     6.52      97.85     97.46     34.86 

336.  6  qt.  +  5  qt.  =  how  many  gal.  and  qt.  ? 

337.  Write  in  Roman  numbers  the  answers  to  the  fol- 
lowing :   Q  +  F^  =  ?     26  +  5=?     36+5  =  ? 

338.  46  pupils  were  in  a  school  and  5  new  pupils  en- 
tered.     How  many  puj^ils  were  in  the  school  then  ? 

339.  Add:  8653     84676     65745     36235    50685    60606 

7527     79515     96186    75486    64506    39545 

How  many  thousands  are  there  in  each  answer  ? 

340.  Write  a  number  that  means  25  thousands,  3  hun- 
dreds, 5  tens,  7  units. 

Write  numbers  made  of  : 

341.  75  thousands,  7  hundreds,  0  tens,  4  units. 

342.  131  thousands,  2  hundreds,  7  tens,  6  units. 

343.  475  thousands,  3  hundreds,  8  tens,  0  units. 

344.  187  thousands,  0  hundreds,  3  tens,  4  units. 

345.  Add  6  to  each  of  ten  numbers  whose  unit  figure 
is  6. 

346.  Add:  8466     19267     83646     96356     72685    96564 

7846     56762    47625     64925     66606    86165 

347.  6  ft.  +  6  ft.  =  how  many  yd.  ? 


ADDITION 


rr 


348.  6  qt.  4-  6  qt.  =  liow  many  gal.  ? 

349.  G  pt.  +  G  pt.  =  how  many  qt.  ?     How  many  gal.  ? 

350.  326 +  G  =  ?   586  +  6  =  ?   916  +  6  =  ?   471  +  0  =  ? 

351.  A  man  paid  8  60  for  a  horse  and  twice  as  much  for 
his  carriage.     How  much  did  the  carriage  cost? 

352.  Add:  $13.76    824.96    818.06    828.76     8128.16 


2.86 


3.36 


1.05       17.06 


25.06 


353.  Copy  Fig.  14  by  placing  inch- 
squares.  Find  the  length  of  the  perim- 
eter of  the  figure  you  have  made.  If 
each  inch  line  were  a  foot  line,  how 
many  yards  long  would  the  perimeter 
be? 


Fig.  14 


354.  Copy  Fig.  15  by  placing 
squares.  Find  the  length  of  the 
perimeter.  How  many  yards  long 
would  the  perimeter  be  if  each  of 
the  inch  lines  were  changed  to 
a  foot  line  ? 

355.  Add  7  to  each  of  ten  num- 
bers that  have  6  for  the  unit  figure. 

356.  Find   the  sum   of    60    and.  Fig.  15 
70.     60  +  50.     436  +  7.     596  + 1 

357.  Add:       8252.75      8187.36      8432.66 

964.27         187.17         719.57 


9  ■    t 


8314.86 
938.97 


358.  6  qt.  +  7  qt.  =  liow  many  gal.  and  qt.? 

359.  6  ft.  +  7  ft.  =  how  many  yd.  and  ft.? 

360.  Write  in  Roman  notation  the  sum  of  15  and  7. 


T8 


ADDITION 


361.  Find  in  the  4th  ten  the  even  nnmber  that  ends  in 
G  and  add  7  to  it. 

362.  In  the  5th  ten  lind  the  even  nnmber  that  ends  in 
8  and  add  7.  ^ 

363.  How  many  days  in  5C)  days  +  1  week? 

364.  Add  8  to  each  nnmber  ending  in  6  and  less  than 
108. 

365.  Write  22  dollars  and  16  cents.  Under  that  write 
31  dollars  and  18  cents.     Add  them.     13.46  +  -^8.28  =  ? 

366.  Write  48  dollars  and  36  cents,  and  add  to  it 
27  dollars  and  38  cents.     $5.08  +  'f  1.96  =  ? 

367.  To  125  dollars  and  58  cents  add  134  dollars  and 
IS  cents. 

368.  16  c[t.  of  milk  and  8  qt.  of  milk  =  how  many  gal. 
of  milk  ? 

369.  6  ft.  +  8  ft.  eqnal  how  mnch  less  than  5  yd.  ? 

370.  Add:  f  1375.58     18934.68     $2456.76     $3467.86 

8277.16        7927.26       7778.18        8881.38 

371.  Add  9  to  each  number  ending  in  6  that  is  less 
than  109. 

372.  60  +  90  =  ?    60  +  80  =  ?    60  +  50  =  ?    60  +  70=? 

373.  6  qt.  +  9  qt.  =  how  many  qt.  ?  How  many  qt. 
must  be  added  to  eqnal  4  gal.? 

374.  16  qt.  +  9  qt.  =  how  many  gal.  and  qt.? 

375.  If  you  measure  off  9  in.  on  a  tape  measure  and 
then  measure  off  6  in.  more,  how  many  ft.  and  in.  will 
you  have  measured  off  ? 

Illustrate  if  necessary. 

376.  Add:    $366.66       $276.76       $346.86        $868.76 

839.19  988.19  997.09  666.09 


ADDITION  7<J 

377.  How  many  pecks  make  a  bushel  ? 

Use  peck  and  bushel  measures  to  show  how  niauy  pecks  equal  a 
bushel. 

378.  HoAv  many  pecks  of  potatoes  in  2  busliels  of 
potatoes?    3  bushels?    4  bushels?    5  bushels?    G  bushels? 

379.  When  potatoes  can  be  bought  for  10  cents  a  peck, 
what  is  the  cost  of  a  bushel  ?     2  bushels  ? 

380.  If  a  bushel  of  potatoes  lasts  a  family  a  week,  how 
many  pecks  will  they  eat  in  2  weeks  ?  What  other 
things  that  we  eat  are  sold  by  the  peck  and  bushel? 

381.  A  pint  equals  what  part  of  a  quart  ?  What  part 
of  a  bushel  equals  a  peck?  A  quart  equals  what  part  of 
a  gallon  ?     A  foot  equals  what  part  of  a  yard  ? 

382.  To  several  numbers  ending  in  5  add  6. 

Ilecall  previous  combinations  and  show  that  many  of  the  follow- 
ing are  the  same,  stated  in  reverse  order. 

383.  50  +  60  =  ?     60  +  70  =  ?     80  +  30  =  ? 

384.  Add:  $312.75        $427.35       $813.25       $875.45 

999.06  897.26  729.36  696.16 

385.  5  qt.  +  6  qt.  =  how  many  gal.  and  qt.? 

386.  15  pk.  +  6  pk.  =  how  many  pk.  ?  How  many 
more  than  5  bu.? 

387.  25  pk.  +6  pk.  =  how  many  pk.  ?  How  many 
pk.  less  than  8  bu.  ? 

388.  AVrite  in  Roman  notation  the  sum  of  .5  and  6.  15 
and  6.     25  and  6. 

389.  Add  7  to  5,  and  to  8  other  numbers  whose  unit 
figure  is  5. 

390.  Add:  $817.35        $428.75        $623.85       $497.35 

219.37  879.27  899.17  895.27 


80 


ADDITIOX 


391.  5  pk.  +  7  pk.  =  how  many  bu.? 

392.  15  pk.  4-  7  pk.  =  how  many  bu.  and  pk.? 

393.  25  pk.  +  7  pk.  =  how  many  bu.? 

394.  15  ft.  +  7  ft.  =  liow  many  yd.  and  ft.? 

395.  (5  +  7)h-2=?      (5  +  7)^3=?      (5  +  7)^4=? 

396.  Write  in  Roman  notation  the  sum  of  15  and  7. 
25  and  7. 

397.  Add  8  to  5  and  to  10  other  numbers  whose  unit 
figure  is  5. 

398.  Add  8  cents  to  11.25.     Add  8  cents  to  17.75. 

399.  Add:  1122.85      11285.75      12775.65       $(348.55 

399.58         4988.18         9888.28  49.78 

400.  John  caught  28  fish  in  the  morning  and  5  in  the 
afternoon.     How  many  fish  did  he  catch  that  day? 

401.  Copy  Fig.  1(3  by  pLacing  triangles 
that  are  each  one  half  of  an  inch  square. 
How  long  is  the  perimeter  of  the  figure  you 
have  made  ? 

402.  Copy  Fig.  16  by  drawing  it,  measur- 
ing your  lines  carefully. 

Let  pupils  draw  to  a  scale. 

403.  Show  1  of  Fig.  16.     Show  i  of  it.     Show  ^  of  it. 

404.  How  many  eighths  equal  |-  of   it?      How  many 

eighths  equal  ^  of  it? 

405.  Copy  Fig.  17  by  placing  tri- 
angles. Show  -|-  of  it.  Show  J  of  it. 
How  many  eighths  in  ^  of  it?  In  \ 
of  it? 

Fig.  17  406.    Copy  Fig.  17  by  drawing. 


Fig.  16 


ADDITION 


81 


407.  Add  9  to  5  and  to  10  other  numbers  whose  unit 
figure  is  5. 

408.  Add:  1369.75       $S24.65       1614.35       1648.15 

975.19  799.09  999.29  396.79 

409.  5  pt.  4-  9  pt.  =  how  many  qt.? 

410.  15  ft.  +  9  ft.  =  how  many  yd.? 

411.  15  pk.  +  9  j)k.  =  how  many  bu.? 

412.  Read  VI,  VIII,  XII,  XV,  XVI,  XXVII. 

413.  L  stands  for  50.     How  much  are  L  and  I  ? 

414.  Read  LXI,  LXIII,  LXV,  LXVII,  LXVI. 

415.  Write  in  Roman  notation  22,  33,  17,  37,  13,  61. 

416.  Add  7  to  4  and  to  all  the  numbers  that  are  less 
than  105  and  end  in  4. 

417.  Add:  1394.84       $687.24         187.34         1672.84 

97.07  97.17  274.37         5998.07 

418.  14  pk.  +  7  pk.  =  ?     How  much  more  than  5  bu.? 

419.  1  gal.  +  7  qt.  =  how  many  qt.  ? 

420.  34  days  -f  1  week  =  how  many  days  ? 

421.  Joseph  had  a  kite  string  64  feet  long.  His  mother 
tied  on  string  enough  to  make  it  71  feet  long.  How  much 
did  she  lengthen  the  kite  string  ? 

422.  Copy  Fig.  18  by  plac- 
ing inch  squares.  How  long  is 
the  perimeter  of  your  figure  ? 

423.  Copy  Fig.  18  by  draw- 
ing.      Make    each   square   J  -p^^  ^g 

inch  each  wav. 

1/ 

424.  ]\Iake  a  list  of  ten  numbers  Avhose  unit  figure  is  4, 
and  add  8  to  each  of  them. 


HORN.    ARITH. 


6 


ADDITION 

Add:  124.64 

1194.34 

$294.74 

$245.54 

78.28 

928.58 

998.08 

98.28 

82 

425. 


426.  4  pt.  of  berries  and  8  pt.  of  berries  are  liow  many  qt.  ? 

427.  14  qt.  -h  8  qt.  =  how  many  gal.  and  qt.  ? 

428.  4  pk.  of  apples  and  8  pk.  of  apples  =  how  many  bu.? 

429.  When  Henry  was  going  to  Boston  on  the  cars  he 
found  at  12  o'clock  that  he  had  ridden  74  miles.  He  rode 
8  miles  more.     How  far  from  his  home  was  Boston  ? 

430.  How  long  is  a  square  whose  perimeter  is  12  inches  ? 
Draw  it.  Divide  it  off  into  square  in.  How  many  are 
there  ? 

431.  Add  9  to  each  of  ten  numbers  whose  unit  figure  is  4. 

432.  To  I)  214.24  add  1 89.39.  To  175.74  add  1129.14. 
To  $317.74  add  $98.99.     To  1361.47  add  $127.91. 

433.  4  pecks  of  apples  were  in  a  bin  and  9  pecks  were 
put  into  the  bin.  How  many  bushels  of  apples  might  be 
taken  from  the  bin,  and  how  many  pecks  would  be  left  ? 

434.  Add  8  to  each  of  ten  numbers  that  end  with  3. 

435.  Add  $234. 35  to  $537.38.  Add  $596. 38  to  $398. 73. 

436.  3  qt.  +2  gal.  =  how  many  qt.  ? 

437.  Write  in  Roman  numbers  the  sum  of  13  and  8  ; 
of  23  and  8. 

438.  Add  9  to  each  of  ten  numbers  that  end  with  3. 

439.  Add  $784.39  to  $20.94.    Add  $64.39  to  $2125.73. 

440.  13  pk.  +  9  pk.  =  how  much  more  than  5  bu.  ? 

441.  Write  12  numbers  that  end  in  2  and  add  9  to 
each  of  them. 

442.  Add  $317.12  to  $14.09.    Add  $923.54  to  $96.49. 

443.  Add  9  to  the  first  even  number  in  the  fourth  ten. 

444.  C   stands  for   100.      What  must  CC  stand   for? 

445.  Read  CX,  CL,  CV,  CI,  CCXV,  CCCXX. 

446.  Write  in  Roman  notation  105,  108,  211,  313. 


CHAPTER 

V 

SUBTRACTION 

Difference,  Minuend,  Subtrahend,  Pound, 

AND    0 

unci 

NUMBER  TABLE 

1         11 

21 

31 

41 

51 

61 

71 

81 

91 

2        12 

22 

32 

42 

52 

62 

72 

82 

92 

3        13 

23 

33 

43 

53 

63 

73 

83 

93 

4        14 

24 

34 

44 

54 

64 

74 

84 

94 

5        15 

25 

35 

45 

55 

65 

75 

85 

95 

6        IG 

26 

36 

46 

56 

66 

76 

86 

96 

7        17 

27 

37 

47 

51 

61 

77 

87 

97 

8        18 

28 

38 

48 

58 

68 

78 

88 

98 

9        19 

29 

39 

49 

59 

69 

79 

89 

99 

10    20 

30 

40 

50 

60 

70 

80 

90 

10( 

1.  Cover  the  last  three  numbers  of  the  first  ten  in  the 
number  table  and  tell  how  many  of  the  ten  remain  in 
sight. 

2.  Cover  the  last  three  numbers  of  the  first  twenty  and 
tell  how  many  of  the  twenty  remain.  How  many  tens 
and  how  many  units  remain  ? 

3.  In  the  same  way  take  2  from  30  and  show  how  many 
units  of  the  broken  ten  remain,  and  how  many  whole  tens. 

83 


84  SUBTRACTION 

4.  Show  liow  many  units  and  how  many  whole  tens 
remain  when  there  is  taken  in  the  same  way  4  from  40,  5 
from  20,  6  from  30,  3  from  70,  2  from  80,  6  from  70,  4 
from  30. 

5.  From   30  Lead  the  children  to  see  that  in  the  written 

take      4      work  they  are  taking  four  from  one  of  the  tens 

—  just  the  same  as  they  did  in  subtracting  on  the 
number  table,  and  that  hence  there  are  only  two  tens  left  to  be  ex- 
pressed in  the  remainder. 

6.  From  90   60   80   50   70   40   30   60 

take  _I_^_i_5_^_^_^_^ 

7.  Find  40  on  the  number  table  and  from  it  subtract 
14,  taking  4  away  first  and  then  10.  Point  out  the  an- 
swer. 

8.  In  the  same  way  subtract  13  from  30,  12  from  50, 
15  from  60,  13  from  70,  14  from  60,  12  from  80. 

9.  From   50  The  children  should  be  led  to  see  that  writ- 

tike   14       ^^^^  subtraction  corresponds  exactly  to  the  proc- 

—  ess  by  which  they  have  subtracted  by  the  help 
of  squares  or  the  number  table,  and  that  it  is  only  a  convenient  way 
of  getting  the  same  results. 

10.  From       60       70       80       40       30       90       70       80 

take       2315472622383541 

11.  When  one  number  is  subtracted  from  another,  the 
result  is  called  their  Difference.  What  is  the  difference 
between  9  and  11  ?  Between  19  and  21  ?  Between  39 
and  41  ?     Between  59  and  61  ? 

12.  Give  the  difference  between  13  and  17,  13  and  27, 
13  and  37,  13  and  47,  13  and  57,  13  and  67,  13  and  77. 

It  is  assumed  that  chart  drill  will  be  given  frequently,  hence  no 
more  chart  exercises  are  given  in  this  chapter, 


SUBTRACTION.  85 

13.  2  from  11  =  ?  9  from  11  =  ?  2  from  21  =  ? 
2  from  71  =  ?            9  from  71  =  ?  9  from  21  =  ? 

14.  Name  all  the  numbers  that  end  in  1  and  are  less 
than  100,  and  subtract  2  from  each  of  them.  Subtract 
9  from  each  of  them. 

15.  Which  is  more,  21  or  10  +  11  ?     51  or  40  +  11  ? 

16.  From    51  Explain  that  as  we  cannot  take  2  units  from 
take      1'^      ^  unit,  we  take  2  units  from  11  units,  and  write 

—  9  units  in  the  answer.  40,  or  4  tens,  are  left, 
and  from  them  we  subtract  the  1  ten,  and  write  the  3  tens  in  the 
answer. 

17.  From     81      71      61      41      51      81      91      31      481 

take    12     12      22      19     29      29      29      29      139 


18.  Kate  had  11  cents  and  spent  all  but  2  of  them. 
How  many  did  she  spend  ? 

19.  Find  some  numbers  that  are  made  of  a  number  of 
tens  and  one  unit,  and  subtract  3  from  each  of  them. 
Subtract  8  from  each  of  them. 

20.  Madge  had  21  cents  and  spent  3  cents  for  a  pencil. 
How  many  cents  had  she  left  ?  She  bought  a  tablet  for  7 
cents.  How  many  cents  were  left  ?  She  gave  all  but  8 
cents  to  her  little  brother.     How  many  cents  did  she  give 


him  ? 

21. 

Find  differences  : 

91 

71         141         781 

961 

31 

5141 

8191 

73 

23          18        343 

438 

18 

328 

4823 

22.  Write  in  Roman  numbers  the  difference  between 
3  and  81. 

23.  Subtract   4   from   each  of  several   numbers  whose 
unit  figure  is  1.     Subtract  7  from  tlie  same  numbers. 

24.  From  281      871      321      961      3481      2191      6111 

take  134      537      114      237      1718      1437      3283 


86 


SUBTRACTION 


25.  Nellie  found  11  eggs  in  a  nest,  but  broke  4  in 
carrying  them  to  the  house.  How  many  eggs  did  she 
bring  in  ? 

Call  for  story  problems. 

26.  Subtract  5  from  each  of  several  numbers  that  end 
in  1.     Subtract  6  from  the  same  numbers. 

27.  From      2141      3171      8111      3191      8171      1819 
take        1614        556      1655      1505      1615        263 


BOARD    WORK 

r2 

6                      3 

7 

Learn   to  sub- 

>om  11 
take 

5 
9 
3 

7 
4 

8 

tract  quickly  each 
of    the    numbers 

5                   11 

9 

on  the  edge  of  the 
square  from  11. 
Then  change    11 

is 

8 

4 

to  21,  31,  etc. 

28.  A  line  lacks  1  inch  of  being  1  foot  long.  If  9 
inches  of  it  were  rubbed  out,  how  long  would  it  be  then  ? 

29.  21  qt.  —  5  qt.  =  how  many  gal.? 

30.  21  pk.  —  9  pk.  =  how  many  bu.? 

31.  Write  in  Roman  notation  the  number  which  is  the 
difference  between  21  and  4  ;  the  difference  between  61 
and  4. 

32.  Copy  Fig.  1  by  placing  triangles  made  by  cutting 

inch- squares  in  two.  Show  J  of 
the  figure.  Show  ^  of  it.  How 
many  eighths  in  |  ? 

33.    Divide    the    figure    into    4 
equal  parts.      How  many  eighths 


Fig.  1 


inf? 


Ill  i  ? 


Ill  i  ? 


SUBTRACTION 


87 


Fig.  3 


Show  I  of  it. 


34.  Copy  Fig.  1  by  drawing. 

35.  Copy  Fig.  2  by  placing  an  inch- 
square  and  triangles  made  by  bisecting 
an  inch-square.  Show  ^  of  the  figure. 
ShoAv  ^  of  it.     How  many  fourtlis  equal  ^  ? 

36.  Copy  Fig.  2  by  drawing. 

37.  Copy  Fig.  3  by  placing  triangles. 
Show  J  of  the  figure.  Show  ^  of  it. 
How  many  sixths  in  -J?  Sliow  J  of  it. 
How  many  sixths  in  J  ?  Show  ^  of  it. 
How  many  sixths  in  |  ? 

38.  Copy  Fig.  3  by  drawing. 

39.  Show  on  the  number  table  J  of  20  ;   |  of  40  ;  J  of 
60  ;  1  of  80  ;   J  of  100. 

40.  Show  1  of  30  ;  f  of  30  ;  i  of  60  ;  |  of  60. 


Pig.  3 


J  of  90  =  ? 


f  of  90  =  ? 


1  of  40  =  ? 


f  of  40  =  ? 


41.    From  numbers  ending  in  2  subtract  3.     From  the 
same  numbers  subtract  9. 


42.    Find  differences  : 

82       92       72       182 
13       43       29         79 


562 


4262 
1319 


8232 
2329 


7212 
2449 


43.  When  butter  is  22  cents  a  pound  and  lard  is  9  cents 
a  pound,  the  price  of  a  pound  of  butter  is  how  much  more 
than  the  price  of  a  pound  of  lard  ? 

44.  From  numbers  ending  in  2  subtract  4.  From  the 
same  numbers  subtract  8. 


45.  Find  differences  : 

272   9262   6224 

8172 

9182 

8272 

9262 

134   7814   2354 

2518 

6814 

2714 

5438 

88  SUBTRACTION 

46.  Eva  spent  22  cents  in  one  day.  She  spent  8  cents 
before  dinner.     How  much  did  she  spend  after  dinner  ? 

47.  From  numbers  ending  in  2  subtract  5.  From  the 
same  numbers  subtract  7. 

48.  Find  differences  : 

131.72    1818.23    1712.92    1921.92    $681.82    1492.62 
6.15       391.51         87.65       238.17       126.15       385.17 

49.  Henry  bought  a  sled  for  $.72  and  traded  it  for 
another  sled  and  a  nickel.  How  much  was  the  other  sled 
worth  ? 

50.  From  numbers  ending  in  2  subtract  6. 

51.  Find  differences  : 

12352    81292    92342    82322    98292    22222 
6146     2536    24128    37156    63526    17516 


From  12  take  .  . 


52.  The  city  where  Alfred  lives  is  22  miles  from  Bos- 
ton. When  he  has  ridden  6  miles  towards  Boston,  how 
far  is  he  from  it  ? 

BOARD   WORK. 

Practice  subtracting 
rapidly  each  number  on 
the  edge  of  the  circle 
from  the  number  at  the 
center.  Then  replace 
32  with  52,  72,  92,  etc. 

53.  22  qt.  —  4  qt.  =  how  many  gal.  ? 

54.  Write  in  Roman  numbers  the  difference  between 
22  and  7. 

55.  Mary  bought  a  dozen  eggs  and  broke  5  eggs  carry- 
ing them  home.     How  many  were  left  ? 

56.  One  dozen  minus  one  half  a  dozen  =  how  many? 


SUBTRACTION  89 

57.  If  9  eggs  were  taken  from  a  nest  where  a  dozen 
eggs   were  found,  liow  many  would  remain? 

58.  The  number  from  which  another  number  is  sub- 
tracted is  called  the  Minuend.  Make  problems,  using  12 
as  a  minuend. 

59.  Use  13  as  a  minuend,  subtracting  4.  Subtract  4 
from  eisfht  other  numbers  that  end  in  3.  From  the  same 
numbers  subtract  9. 

60.  Read  minuends  and  find  differences  : 

8373        2383        9323        4393        6373        33333       7533 
3439  924        5439        2489        2429        21919       3214 

61.  13  qt.  of  berries  — 9  qt.  of  berries  =  how  many  gal.  ? 

62.  From  several  numbers  ending  in  3  subtract  5. 
From  the  same  numbers  subtract  8. 

63.  Find  differences  : 

7343   4283   6293   8083   9639   33333   8343 
985    948   1745   4728   6255   18175   2519 

64.  (13-5)^2  =  ?   (23-5)-^2  =  ?   (12-6)-3  =  ? 
(22-7)^3  =  ?   (13-5) -4  =  ?   (33-6) +4  =  ? 

65.  If  a  piece  of  ribbon  5  ft.  long  is  cut  from  a  piece 
4  yards  and  1  foot  long,  how  much  ribbon  is  left  ? 

66.  Write  in  Roman  notation  the  difference  between 
73  and  5  ;   between  33  and  5. 

67.  From  numbers  endinsr  in  3  subtract  6.  From  the 
same  numbers  subtract  7. 

68.  823  -  456  =  ?  733  -  276  =  ?  1039  -  462  =  ? 

69.  23  pk.  —  7  pk.  =  hoAv  many  bu.? 
See  board  work  used  with  numbers  11  and  12. 

70.  Write  in  Roman  numbers  tlie  difference  between 
7  and  33. 


90  SUBTRACTION 

71.  23  qt.  —  7  qt.  =  how  many  gal.? 

72.  33  pk.  —  9  pk.  =  bow  many  bu.? 

73.  23  ft.  —  5  ft.  =  how  many  yd.? 

74.  John  had  13  cents  and  spent  8  of  them.  How  many 
were  left  ? 

75.  13  Readers  belong  to  the  library  and  6  are  in  use. 
How  many  are  on  the  shelves  ? 

76.  Make  problems  in  which  j^ou  use  13  as  a  minuend. 

77.  Read  XV,  XIII,  C,  CVI,  CX,  CXIII,  CXXII. 

78.  IV  means  4.     Read  XIV,  XXIV,  XXXIV,  LXIV. 

Show  that  when  the  smaller  Roman  numeral  is  written  before  the 
larger,  their  difference  is  expressed. 

79.  What  does  IX  mean  ?  How  can  you  tell  ?  Read 
XIX,  XXIX,  LIX,  LXXIX,  XXXIX. 

80.  Write  in  Roman  numbers,  59,  89,  39,  14,  64,  74,  24. 

81.  From  14,  and  from  some  other  numbers  whose  unit 
figure  is  4,  subtract  5.  Subtract  9  from  the  same  num- 
bers. 

82.  Find  differences  : 

9484  8474        7494        2094        6434        7494        8484 

6935  3529        _935        1829        1875        2965        2569 

83.  Of  14  horses  hauling  loads,  5  were  white.  Hoav 
many  of  them  were  not  white  liorses  ? 

84.  Write  in  Roman  notation  the  number  that  is  the 
difference  between  14  and  5  ;  between  24  and  9. 

85.  2  weeks  —  5  days  =  how  many  days  ? 

86.  From  several  numbers  Avhose  unit  figure  is  4  sub- 
tract 6.     From  the  same  numbers  subtract  8. 

87.  Find  diiferences  : 

2494   3484   7474   9484   11464   14846   9234 
1866   1628   2628   7866     628    8265   4968 


SUBTRACTION  91 

88.  8  yd.  —  6  ft.  =  how  many  ft.?     How  many  yd.? 

89.  6  gal.  —  6  qt.  =  how  many  qt.? 

90.  Write  the  difference  between  XXXIV  and  VIII. 

91.  From  several  numbers  whose  unit  figure  is  4 
take  7. 

92.  Find  differences  : 

324     674     924     1246     6549     846     7434     6434     3444 
207     538     619     Jl62     2275     193     2918     2726     1968 

93.  Edwin  had  24  marbles  and  lost  all  but  7.  How 
many  did  he  lose  ? 

Use  board  work  as  with  numbers  11  and  12. 

94.  Write  in  Roman  numbers  the  number  that  is  the 
difference  between  14  and  8  ;  between  24  and  8  ;  between 
34  and  6. 

95.  A  hen  sat  upon  1  dozen  and  2  eggs.  5  eggs  failed 
to  hatch.     How  many  chickens  came  out  ? 

96.  If  a  room  is  24  ft.  long  and  9  ft.  wide,  how  many 
ft.  greater  is  its  length  than  its  width  ?    How  many  yd.? 

97.  If  you  measure  off  a  line  14  feet  long  on  the  floor, 
and  another  line  8  feet  long,  what  is  the  difference  be- 
tween them  in  feet  ?     In  yards  ? 

98.  Make  problems  using  14  as  a  minuend. 

99.  Read    X,    I,   V,    L,   C,   IV,   IX,   XIX,   XXIX. 

100.  X  written  before  L  means  40.     Can  you  tell  why  ? 
Read   XL,  XLI,  XLII,  XLIII,  XLIX,  CL,  CXL. 

101.  Write  in  Roman  notation  the  difference  between 
40  and  9  ;  4  and  40. 

102.  From  15,  and  other  numbers  whose  luiit  figure  is  5, 
subtract  6.     From  the  same  numbers  subtract  9. 


92  SUBTRACTION 

103.    Find  differences  : 

975    T595    6935    8357    7358    4595 
256    4976    2486    2994    2466    1936 


104.  Joseph  had  Q'^  feet  of  kite  string,  and  his  mother 
used  6  feet  of  it  to  tie  up  a  bundle.  How  many  feet  of 
string  had  he  left  ? 

105.  Express  in  Roman  numbers  the  difference  between 
15  and  6. 

106.  From  15,  and  other  numbers  ending  in  5,  take  7. 
From  the  same  numbers  take  8. 

107.  Find  differences  : 

175  735  8595  1125  3525  6352  8958  7257  9859 
128  219   867  1016  2778  2161  6274  2164  3694 

108.  15  qt.  —  7  qt.  =  how  many  gal.? 

109.  25  pk.  —  2  bu.  =  how  many  pk.? 

110.  In  Roman  numbers  write  the  difference  between 
7  and  35 ;  between  45  and  8. 

111.  If  I  saw  off  8  in.  from  a  board  that  is  1  ft.  and 
3  in.  long,  how  many  in.  are  left  ? 

112.  355  is  a  minuend  and  127  the  number  to  be  sub- 
tracted.    What  is  the  difference  ? 

113.  A  number  which  is  subtracted  from  another  num- 
ber is  called  a  Subtrahend.  Make  a  problem  with  9  as  a 
subtrahend. 

114.  Make  problems  in  wliich  you  use  15  as  a  minuend, 
and  some  number  less  than  10  for  a  subtrahend. 

115.  Subtract  7  from  16,  and  from  other  numbers  end- 
ing in  6.     From  the  same  numbers  subtract  9. 

116.  Find  differences  : 

13.76     15.76     128.26     19.46    118.68    19.36 
Subtrahend  2.37       2.49  9.87       2.59       13.75       4.98 


SUBTRACTION  93 

117.  Write  in  Roman  notation  the  difference  between 
56  and  9.     Between  66  and  7. 

118.  Anna  is  16  years  old  and  Mary  is  9  years  old. 
How  much  older  is  Anna  than  Mary  ? 

119.  William  is  16  years  old  and  Thomas  is  9  years 
younger.     How  old  is  Thomas  ? 

120.  I  have  a  string  that  is  1  ft.  and  4  in.  long.  If  I 
break  off  a  piece  7  in.  long,  how  much  will  remain  ? 

121.  Anna  expected  to  spend  26  days  in  visiting  a 
friend,  but  was  called  home  a  week  sooner  than  she 
expected.     How  long  did  she  stay  ? 

122.  HoAV  many  ounces  make  a  pound  ? 

Let  the  children  weigh  out  sand,  sawdust,  coal,  or  some  other  sub- 
stance until  they  realize  the  meaning  of  the  terms  "pound"  and 
"  ounce." 

123.  How  many  ounces  in  1  pound  lacking  7  ounces  ? 
1  pound  lacking  9  ounces  ? 

124.  At  10  cents  a  pound,  how  many  pounds  of  candy 
can  be  bought  for  50  cents  ? 

125.  J  a  pound  of  sand  weighs  how  many  ounces  ?  |^  a 
pound  of  sugar  Aveighs  how  many  ounces  ? 

126.  Use  8  as  a  subtrahend  with  each  number  less  than 
100  whose  unit  figure  is  6, 

127.  Find  differences  : 

116.16     126.36     i278.36     8376.46     1236.56     1376.86 
8.08  8.18  54.28  18.28  98.38         98.78 

128.  Make  problems  using  16  as  a  minuend,  and  a 
smaller  number  as  a  subtrahend. 

129.  Use  9  as  a  subtrahend  with  each  of  the  numbers 
less  than  120  whose  unit  figure  is  7. 


94  SUBTRACTION 

130.  What  number  added  to  39  makes  47?  What 
number  added  to  339  makes  347  ? 

131.  Find  differences  : 

1377.57        1627.87        1547.27        12275.87        1648.77 
88.29  38.39  68.49  999.38  294.89 

132.  Anna  had  a  dime  and  7  cents  and  bought  an  8-cent 
doll.     How  much  had  she  left  ? 

133.  A  flower  bed  is  27  feet  long,  and  John  has  weeded 
9  feet  of  it.     How  much  remains  to  be  weeded  ? 

134.  Make  problems  using  17  as  a  minuend. 

135.  XC  means  90.     Can  you  tell  why  ? 

136.  Read  XCI,  XCV,  XCIV,  XCIX,  XCVI,  XCIH, 
XCVIII. 

137.  In  Roman  numbers  write  : 

All  tlie  numbers  that  end  in  9  up  to  99.  All  the  num- 
bers that  end  in  4  and  are  less  than  100.  Your  age.  All 
the  even  numbers  in  the  first  two  tens. 

138.  Subtract  9  from  several  numbers  ending  in  8. 
139. 

Minuends  727     113.47     167.27      167.75     $98.75 

Subtrahends        259         9.18        29.18        38.69        26.97 
Differences  ? 


CHAPTER   VI 


APPLICATIONS   OF   ADDITION   AND    SUBTRACTION 

Industrial  Problems,  Days  in  Months,  Odd  Numbers 

1.  Add  729  to  itself.     Add  1348  to  itself. 

2.  Find  the   sum   of   648   and  the   number  tliat  is  1 
S^ieater  than  648. 

3.  Find  the  sum  of  276  and  the  number  that  is  1  less 
tlian  276. 

4.  Add  7  times  2  to  45.     (2  x  8) +  (10  x  7)=  ? 

5.  Add  6  times  2  to  the  6th  multiple  of  10. 

6.  What  must  be  added  to  9  to  equal  12?    17?    15? 

7.  26-?  =  13.     64-?  =  58.     35-?  =  27.     26-?  =  17. 

8.  Copy  Fig.  1  by  drawing  3  inch-squares 
and  bisecting  them. 

9.  Show  1  of  the  figure.  Show  |  of  it. 
Show  -^  of  it.  How  many  6ths  does  ^  e(|ual  ? 
Show  ^  of  it.     How  many  6ths  in  J  ? 


Fig.  1 


10. 
11. 
12. 
13. 
14. 


6.  _  1  _    ? 
6  6   ~"  6 


5   _  1  _   ? 

"6  2   "~  6' 


5.  _  i  —  1  2. 

6  3  ~"  6*  3 


1  —  1 
6   "  6* 


Show  on  the  number  table  ^  of  50.     -|  of  50.     |-  of  50. 
Show  1  of  60.     I  of  60.     I  of  40.     f  of  40. 
Add  I  of  50  to  50.     To  25.     To  30.     To  20. 
Add  1  of  60  to  60.     To  30.     To  40.     To  25. 


15.    Subtract  i  of  30  from  30.     J  of  30  from  75. 

95 


96       APPLICATIONS   OF   ADDITION   AND   SUBTRACTION 

16.  Subtract  i  of  40  from  40.     ^  of  40  from  60. 

17.  Find  the  difference  between  464  and  820.  Which 
number  is  the  minuend  ? 

18.  Find  the  difference  between  398  and  785.  Which 
is  the  greater  number,  the  minuend  or  subtrahend  ? 

19.  Find  the  difference  between  4246  and  3278.  Where 
is  the  minuend  written  in  subtraction  ? 

20.  484  is  how  many  more  than  376  ?  Which  number 
is  the  subtrahend  ? 

21.  324  is  how  many  less  than  486  ? 

22.  How  many  children  are  there  in  a  ward  school 
which  has  139  children  in  the  first  grade,  137  in  the  2d, 
747  in  the  3d,  128  in  the  4th,  98  in  the  5th,  83  in  the  6th, 
77  in  the  7th,  and  48  in  the  8th? 

23.  158  children  were  in  the  first  grade  of  a  school,  and 
43  were  transferred  to  another  building.  How  many  re- 
mained ? 

24.  There  were  676  children  in  a  school  building  when 
183  others  were  transferred  to  it.  How  many  were  there 
then  ? 

25.  There  were  392  books  in  the  school  library,  and  219 
new  ones  were  added.  How  many  were  in  the  library 
then  ? 

26.  18,943  bushels  of  coal  were  dug  from  a  mine  in  one 
week  and  29,312  the  next  Aveek.  How  many  in  the  two 
weeks  ? 

Take  up  the  subject  of  coal  mining,  showing  coal  and  pictures  of 
mines,  and  reading  or  telUng  stories  about  mines  and  miners.  Then 
let  the  children  give  problems  about  them. 

In  the  same  way  deal  with  the  different  industries  referred  to  in 
the  following  problems,  letting  the  children  furnish  facts  when  they 
can  about  industries  of  which  they  have  some  knowledge. 


APPLICAIIOXS   OF   ADDiriON   AND   SUBTRACTION       97 

27.  18,946  bushels  of  coal  were  dug  from  a  mine  in  one 
week,  29,321  the  next  week,  and  31,457  the  next  week. 
How  many  were  dug  out  in  the  three  weeks  ? 

28.  7281  cattle  were  on  a  cattle  ranch  and  943  were 
killed.     How  many  were  left  ? 

29.  An  iron  foundry  made  875  stoves  in  one  Aveek,  873 
in  another  week,  and  884  in  another  week.  Hoav  many 
in  all  ? 

30.  A  cotton  mill  wove  10,87(3  yd.  of  cloth  in  one  week, 
9343  in  the  next  week,  and  11,833  in  the  next  week.  How 
many  in  the  three  weeks  ? 

31.  A  farmer  raised  2343  bu.  of  corn  in  one  year,  3124 
in  another  year,  1957  in  another  year,  and  2417  in  another 
year.     How  many  did  he  raise  in  those  four  years  ? 

32.  One  farmer  raised  1247  bu.  of  wdieat,  another  far- 
mer raised  3268  bu.,  and  another  farmer  raised  5324  bu. 
How  many  bushels  of  wheat  did  they  all  raise  ? 

33.  A  lawyer  earned  $  5727  in  one  year,  $  2938  in  the 
next  year,  and  'ff  11,536  in  the  third  year.  How  many 
dollars  did  he  earn  in  the  three  years? 

34.  Gold  worth  12,342  dollars  was  taken  from  a  gold 
mine  in  one  month,  98,676  dollars'  worth  in  the  next 
month,  and  321  dollars'  worth  in  the  next  month.  How 
many  dollars  were  taken  out  in  the  three  months  ? 

35.  A  farmer  sent  to  market  one  year  1224  pounds  of 
butter,  1376  pounds  the  next  year.  1312  pounds  the  next 
year,  and  1678  pounds  the  next  year.  How  many  pounds 
of  butter  did  he  send  in  the  four  years  ? 

36.  A  milkman  sold  943  quarts  of  milk  in  Jan.,  836  in 
Feb.,  972  in  Mar.,  and  937  in  Apr.  How  man}^  quarts 
did  he  sell  in  all  ? 

HORN.     ARITH.  7 


98        Ai'l'LlCAllOxNS    OF    ADDITIONS'    AND    iSUBTKACTlOxN 


37.  Copy  and  learn  : 

Thirty  days  hath  September,  April,  June,  and  November. 

All  the  rest  have  thirty-one,  except  February  alone, 

Which  has  just  twenty-eight  in  fine,  till  leap  year  gives  it  twenty-nine. 

Let  children  consult  calendar. 

38.  Copy,  writing  the  number  of  days  in  each  month 
opposite  its  name  : 

March  [  June 

Spking  ■  April  Summer]  July 

May  [  August 


Fall 


Sej^tember 

October 

November 


December 
WiNTEK  ]  January 


February 

39.  How  many  days  in  the  spring  months  ?  In  the 
summer  months?  In  the  fall  months?  In  the  winter 
months  ? 

40.  How  many  days  in  the  last  5  months  of  the  year  ? 
In  this  month  and  last  month  ? 

41.  The  year  in  which  February  has  29  days  is  called 
leap  year,  and  comes  once  in  4  years.  A  boy  named 
Walter  Jones  was  born  February  29th,  1884.  In  what 
year  can  he  first  celebrate  his  birthday  on  the  29th  of 
February  ? 

42.  Find  the  number  of  days  in  the  month  in  which 
you  were  born,  add  to  it  the  number  in  the  montli  before 
and  the  month  after. 

43.  Find  the  number  of  days  in  the  montli  in  Avhich 
Christmas  comes,  add  to  it  the  days  in  the  month  before 
and  the  month  after. 

44.  How  many  days  in  the  first  ten  months  of  a  leap 
year  ? 


APPLICATIONS   OF   ADDITION   AND   SUBTRACTION       99 

45.  Add  the  days  in  the  month  in  which  Thanksgiving 
comes  to  those  in  the  month  after  and  the  month  before. 

46.  A  store  sold  927  yards  of  carpet  in  one  day,  713 
the  next,  and  837  the  next.     How  many  in  all  ? 

47.  Susan's  father  earned  iloOO  in  one  year  and  spent 
11321.     How  much  did  lie  save  ? 

48.  John  gets  2  cents  a  quart  for  picking  herries  for  a 
farmer  and  1  cent  a  quart  for  selling  them.  How  many 
cents  did  he  earn  in  the  day  in  which  he  picked  12  quarts 
and  sold  10  of  them  ? 

49.  A  bookkeeper  earned  f  1400  in  one  year  and  saved 
f  227.     How  much  did  he  spend? 

50.  A  farmer  raised  2827  busiiels  of  corn  ;  another 
farmer  raised  3431,  another  9852,  and  another  6856. 
How  many  bushels  did  all  raise  ? 

51.  Mr.  Smith's  salary  was  11300  last  year  and  11450 
this  3^ear.  How  much  more  does  he  receive  this  year 
than  last  year  ? 

52.  ]\Ir.  Ward  has  3  horses.  Black  Beauty,  Whiteface, 
and  Fleet.  Black  Beauty  is  valued  at  ^375,  Whiteface 
at  1125,  and  Fleet  at  ^575.  How  much  are  they  all 
worth  ? 

53.  Anna  worked  73  problems  in  addition  in  1  week, 
her  sister  worked  98,  and  her  brother  113.  How  many 
did  they  all  work  ? 

54.  John  had  a  knife  worth  49  cents,  w^hich  he  traded  for 
William's  knife  and  2  nickels.  How  much  was  William's 
knife  worth  ? 

55.  Find  the  sum  of  all  the  even  numbers  in  the  first 
ten. 


100     APPLICATIONS   OF   ADDITION   AND    SUBTRACTION 

56.  Numbers  which  are  not  even  are  called  Odd  Num- 
bers. Write  all  the  odd  numbers  in  the  first  ten.  Find 
their  sum. 

57.  Write  all  the  odd  numbers  in  the  second  ten. 
Find  their  sum. 

58.  Find  the  sum  of  the  odd  number  which  comes  just 
before  30  and  the  odd  number  which  comes  just  after  30. 

59.  Draw  a  horizontal  line  9  inches  long,  and  divide 
into  halves. 

60.  Draw  a  vertical  line  5  inches  long,  and  find  how 
many  inches  in  ^  of  it. 


61.  Copy  Fig.  2  by  drawing  inch- 
squares.  Divide  the  figure  into  two 
equal  parts  by  one  straight  line. 

How  many  inch-squares  in  each  half  ? 
Fig.  2 

62.  Can  you  divide  a  group  of  9  children  into  2  equal 
groups  ? 

63.  Can  you  divide  7  apples  equally  between  two  boys 
without  cutting  any  apples  into  halves  ? 

64.  Can  you  find  a  whole  number  that  is  just   |  of 
11? 

65.  Think  of  different  odd  numbers,  and  see  if  you  can 
find  a  whole  number  that  is  just  |-  of  any  of  them. 

66.  Think  of  some  even  numbers,  and  tell  what  -|  of 
each  of  them  is. 

67.  How  are  even  numbers  different  from  odd  num- 
bers? 

68.  Make  a  list  of  the  odd  numbers  in  the  first  two 
tens,  and  find  their  sum. 


1  1      '  '       '         '        >  J  5     ^    > 


APPLICATIONS   OF   ADDITION   AND  tSlTP VfiAcrt;(!)N    !liAl»  ','  ' 

69.  Make  a  list  of  the  even  iiumljers  in  tlie  first  two 
tens,  and  find  their  snni.* 

70.  Mary  was  in  school  20  days  in  the  month  of  Jan- 
uary.    How  many  days  was  she  out  of  school  ? 

71.  Use  23487  as  a  minuend  and  14798  as  a  subtraliend. 

72.  When  one  number  is  subtracted  from  another,  some- 
times the  difference  is  called  the  Remainder.  Find  the 
remainder  Avhen  2987  is  subtracted  from  8012. 

73.  Arthur  had  $38.72  and  spent  I29.8G.  How  much 
was  the  remainder  ? 

74.  Alfred  took  $12.38  from  $21.75.  How  much  was 
the  remainder  ?  ^ 

75.  Find  the  last  remainder  when  from  $829.75  there 
is  subtracted  first  $28.93,  then  $478.38  from  what  was 
left,  then  $312.69  from  what  was  left. 

76.  Subtract  9  from  50  and  write  the  remainder  in 
Roman  numbers.  ^ 

77.  Read  C,  CX,  CL,  CI,  CTH,  CIX,  CXL,  CXLIII, 
CXX,  CCXXV,  (  CCXV,  CCCCLX,  CXLIX. 

78.  Write  in  Roman  notation  all  the  numbers  of  two 
places  that  have  9  in  the  tens'  place  ;  all  the  numbers  of 
two  places  that  have  9  in  the  units'  place. 

*  The  game  of  Odd  or  Even  is  useful  at  this  stage.  Having  the  class 
at  the  board,  the  teacher,  or  the  child  leader,  holds  out  her  closed  hand, 
containing  a  number  of  objects,  — grains  of  corn  or  pieces  of  paper.  Each 
child  writes  "odd"  or  "even."  When  the  hand  is  opened,  those  who 
guess  correctly  credit  themselves  with  the  number,  the  others  with  0. 
After  five  trials  the  scores  are  added.  If  instead  of  using  objects,  the 
leader  simply  writes  a  number  on  paper,  large  numbers  can  be  conven- 
iently used,  and  the  game  thus  varied. 


j^Oii    Ai'.PiLlCA:^,tUNi»>  OF   ADDITION  AND   SUBTRACTION 


79.  What  page  of  your  book  are  you  reading?  The 
numbers  which  show  the  pages  are  written  in  Arabic  nota- 
tion. In  which  kind  of  notation  are  the  numbers  in  tlie 
number  table  written  ? 

80.  Write  in  Arabic  notation  CXC^V,  CCXCI, 
CCCXCVI,  CCCCXCIV. 

81.  Write  in  Arabic  notation,  and  add,  CLXXV,  l^IV, 
LXXXIV,  LXTX,  XLIX,  CXLVJll. 

82.  Subtract  4  from  91  and  write  the  remainder  in 
Roman  numbers. 

83.  Write  in  Arabic  notation  CCXCV  and  CCCLXXVI 
and  find  their  difference. 

84.  C'o}>y  Fig.  o  by  phicing  triangles 
made  by  bisecting  iiicli-squares.  Show 
■1  of  your  ligure.      |  ~  i  =  • 

85.  Divide  the  liij^ure  into  halves. 
How  many  sixtlis  in  one  half  ? 

86.  Copy  Fig.  8  by  drawing. 

87.  Copy  Fig.  4  by  placing  triangles. 
How  many  triangles  does  it  take  ?  How 
many  triangles  would  it  take  to  make 
five  such  figures  ?  To  make  7  such 
figures  ?     To  make  10  such  figures  ? 


Fig.  3 


88.    Show  Jq  of  your  figure. 

1 0  _    1    _  y     J) 3_  _  V      _6^  _ 

10         10  ~"  •       10         10  "~  •        lIT 

4 
10 

Fig.  4 


89. 


Divide   the    fiofure    into    halves. 
How  many  lOths  in  |  ? 

90.    Can  you  take  4  triangles  away  from   Fig.   4  and 
leave  it  just  like  Fig.  3  ?     Copy  Fig.  4  by  drawing. 


APPLICATIONS   OF   ADDITION  AND   SUBTRACTION      103 


Fig.  5 


91.  Copy  Fig.  5  by  placing  triangles.     How  many  tri- 
angles does  it  take  ?    How  many  trian- 
gles would  it  take  to  make  9  such  fig- 
ures?    To  make  6  such  figures? 

92.  Divide  the  figure  into  halves  and 
show  how  many  tenths  in  ^. 

93.  Can  you  take  away  4  triangles 
from  the  figure  so  as  to  make  a  figure 
just  like  Fig.  3  ? 

94.  Can  you  show  how  Fig.  5  can  be  made  just  like 
Fig.  4  by  turning  four  of  the  triangles  around  ?  Copy 
Fig.  5  by  drawing. 

95.  Subtract  327  from  982.  Subtract  it  ao-ain  from 
tlie  remainder  and  again  from  the  second  remainder  and 
see  if  3"our  answer  is  1. 

96.  Keep  on  subtracting  224's  from  1123  in  the  same 
way  until  the  remainder  is  3. 

97.  Subtract  123's  from  369  until  nothing  remains. 
How  many  123's  does  it  take  to  equal  369  ? 

98.  Add  32's  together  until  you  get  192. 
32's  did  you  use  ? 

99.  Add  24's  together  until  you  get  144. 
24's  in  144  ? 


How  many 
How  many 


100.  AVrite  the  names  of  the  months  that  have  31  days, 
^nd  find  how  many  days  in  all  of  them. 

101.  Five   vertical  lines   are   drawn   on  the  board  one 
foot  apart.     How  far  apart  are  the  two  outside  lines  ? 

Let  the  children  try  imagining  before  iUustratiiig. 

102.  If  it  costs  10  cents  to  saw  a  log  into  two  pieces, 
how  much  will  it  cost  to  saw  it  into  three  pieces? 


CHAPTER   VII 

FIVES 

Equilateral  Triangles,  Eoman  Numerals  D  and  M, 

Quotient 

NUMBER  TABLE* 


1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22' 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

65 

75 

85 

95 

6 

16 

26 

36 

46 

r,(cy 

Gij 

76 

86 

96 

7 

17 

27 

37 

47 

51 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10    20    30    40    50   GO   70   80  90  100 

1.  Begin  with  5  and  count  l)y  fives  to  100 
Let  pupils  practice  this  until  they  can  count  rapidly. 

2.  The  number  table  is  divided  into  groups  of  five 
numbers.     Name  all  the   numbers   in  the  first  group  of 

*  Charts  containing  this  and  other  number  tables  should  remain  upon 
the  wall  in  sight  of  the  children  all  the  time,  except  when  tests  are  given. 
By  this  means  the  children  unconsciously  become  familiar  with  the  mul- 
tiples and  their  relative  positions, 

104 


FIVES  105 

five.     Name  all  the  numbers  of   the  second   five.     The 
third  five.     The  fourth  five.     The  fifth  five. 

3.  Show  the  sixth  five.     Show  the  next  five.     Which 
five  is  it  ? 

4.  Point  out  the  second  five  and  name  the  first  and 
last  number  of  it. 

5.  Show  the  third  five  and  name  the  first  and  last 
number. 

6.  What  number  in  the  third  five  is  next  to  the  last  ? 

7.  What  is  the  last  number  of  the  fourth  five  ?     Of 
the  fifth  five  ?     Of  the  sixth  five  ?     Of  the  seventh  five  ? 

8.  In  which  five  is  13  ?    29  ?    31  ?    46  ? 

9.  Name  an  odd  number  in  the  seventh  five. 

Let  the  children  select  numbers  and  tell  in  which  five  they  are 
found. 

10.  Name  in  order  the  multiples  of  five  up  to  100. 
l^earn  to  name  them  without  looking  at  the  number  table. 

Fill  out  and  learn  the  following  table  of  fives  : 

1  five    =  5  fives  =  9  fives  = 

2  fives  =  6  fives  =  10  fives  = 

3  fives  =  7  fives  =  11  fives  = 

4  fives  =  8  fives  =  12  fives  = 

Pupils  must  first  learn  the  nniltiplication  tables  in  regular  order 
so  that  they  may  see  the  aggregations  of  which  multiples  are  com- 
posed. But  in  later  work  care  should  be  taken  not  to  use  a  fixed 
order.  The  child  should  learn  the  statements  of  the  multiplication 
tables  as  separate  facts,  so  that  eacli  may  spring  singly  into  his  con- 
sciousness when  needed. 

11.  Name  the  second  multiple  of  5 ;  the  fourth  multiple 
of  5  ;  the  fifth,  the  sixth,  the  tenth,  the  eighth,  the  seventh. 

12.  3  fives  =  how  many?  What  is  ^  of  15?  5  is  ^  of 
what  number? 


100  FIVES 

13.  5  is  i  of  what  number  ?  How  can  you  tell  ?  5  is 
'I  of  what?  I  of  what?  ^  of  what?  ^2"  ^^  what?  ^  of 
what?     -^^  of  what?     ^  of  what?     -^j  of  what? 

14.  50  is  which  multiple  of  5?  Name  another  number 
that  50  is  a  multiple  of. 

15.  10  is  Avhich  multiple  of  5?  Name  another  number 
that  10  is  a  multiple  of. 

16.  Which  multiple  of  5  is  15?     25?     35?     40?     60? 

17.  Name  multiples  of  5  and  tell  quickly  which  multi- 
ples they  are. 

18.  Name  all  the  numbers  in  the  6th  group  of  five  and 
tell  wiiich  is  the  middle  number. 

19.  5  cents  +  5  cents  +  5  cents  +  5  cents  =  liow  many 
dimes  ? 

20.  5  is  wliat  part  of  25?     Of  40?     20?     60?     35? 

21.  Alary  had  15  cents,  and  Anna  had  J  as  many.  How 
many  had  Anna? 

22.  How  much  is  |  of  15  cents?     |-  of  30  cents? 

23.  How  many  times  can  a  line  5  inches  long  be  meas- 
ured off  on  a  line  20  inches  long?  On  a  line  15  in.  long? 
On  a  line  35  in.  long?     On  a  line  45  in.  long? 

24.  5  multiplied  by  3  =  ?     F>  x  C>  =?     5x8=? 

25.  What  number  is  3  more  than  4  fives  ?  2  more  than 
5  fives? 

26.  How  much  is  1  less  than  3  fives?  2  less  tlian  5 
fives  ? 

27.  Which  is  more,  17  or  3  fives?     How  much  more? 

28.  Wliich  is  more  and  how  much  more,  4  fives  or  18? 
2  fives  or  14?     4  lives  or  23? 

29.  Wliich  is  more  and  how  much  more,  6  fives  or  28? 
4  fives  or  2  tens?     10  fives  or  5  tens?     5  fives  or  26? 


FIVES 


107 


30.  How  many  nickels  equal  15  cents  ?     20  cents  ? 

31.  If  6  little   girls  have   a  nickel  apiece,  how  many 
cents'  worth  of  peaches  can  they  all  buy  ? 

32.  How  many  cents  will  it  cost  for  7  children  to  ride 
on  a  street  car,  if  they  each  pay  5  cents  fare  ? 

33.  Point  out  the  last  number  of  the  lifth  five,  add  2 
fives,  and  point  out  the  result. 

34.  Add  2  fives  to  30  and  point  out  tlie  result.     How 
many  fives  does  it  equal  ? 

35.  Add  2  fives  to  40.     To  50.     35.     45.     55.     15. 

36.  Add  3  fives  to  10.     20.     15.     35.     45.     30.     40. 

37.  Mary    may    think    of    an    even    number    and   tell 
wliich  five  it  is  in.     The  class  may  guess  the  number. 

38.  John  may  think  of  an  odd  number  and  tell  which 
five  it  is  in. 

39.  Begin  at  100  and  c(Hint  backwards  by  fives  quickly. 

40.  Take  2  fives  from  20.     45.     35.     40.     50.     30. 

41.  How  many  fives  in  15?     30  ?     40  ?     55  ?     35  ? 

42.  How  many  fives  in  a  ten  and  half  a  ten  ?    In  2  tens 
and  a  lialf  ?     In  3  tens  and  J  a  ten? 

43.  Place  two  rows  of  five  squares  each  as  in  this 
figure,  and  tell  how  many 
squares  there  are.  Place 
another  row  of  five  squares 
above  them  and  tell  how 
many  squares  there  are.  Keep 
on  placing  rows  of  5  squares 

each  until  the  figure  is  as  wide  as  it  is  long  or  until  it  is 
square.  How  many  little  squares  are  there  in  it  then? 
Find  the  middle  square  of  all  and  write  M  in  it. 

44.  25^5  =  ?     40-5  =  ?     15^5  =  ?     35^5  =  ? 


108 


FIVES 


45.  Read  XC,  XCIV,  CCLII,  CCCX,  CCCLXVI. 

46.  D.  stands  for  500  in  Roman  notation.      Read  DC, 
DL,  DXC,  DCCC,  DXLVIII,  DCLXVI,  DCCIX. 

47.  Write      in      Arabic      notation      DCCXXV      and 
DCCCXXXVII.     Then  find  their  sum. 

48.  Write  in  Roman  notation  605,  607,  609,  611o 


BLACKBOARD    EXERCISE 


Mnltij^ly  5  by  each  of  the 
numbers  on  the  edge  of  the 
triangle.     Answer  quickly. 


Fig.  1 


10  5  9 

49.    Triangles    whose    sides    are    all    equal   like 
Fig.  1  are  called  Equilateral  Triangles.     Are  those 
triangles  equilateral  that  are  made   by  cutting  a 
square  inch  into  halves  ? 
50.    If  each  side  of  Fig.  1  were  5  in.  long,  how  long 
would  the  perimeter  be  ? 

Equilateral  triangles  should  be  furnished  for  the  following  work. 

51.  Copy  Fig.  2  by  placing  equilateral 
triangles.  If  each  side  of  the  triangles 
you  used  were  5  in.  long,  how  long 
would  the  perimeter  of  your  figure  be  ? 

52.  Copy  Fig.  3  by  placing  equilateral 
triangles.  If  each  side  of  the  triangles 
were  5  in.  long,  Avhat  would  be  the 
length  of  a  line  that  would  lie  all  around 


Fig.  3 


Fig.  3 


the  figure? 


FIVES  109 

53.  Show  f  of  Fig.  3.     ShoAv  -|  of  it.     f  -  f  =  ? 

54.  Draw  a  vertical  line  5  in.  long.  Divide  into  inches 
and  mark  the  divisions.  One  in.  is  what  part  of  5  in.? 
3  in.  are  what  part  of  5  in.?     4  in.  are  what  part  of  5  in.? 

55.  How  mucli  do  |  of  the  line  lack  of  being  the  whole 
line  ?     Show  |  of  the  5-inch  line. 

56.  Draw  a  line  |  as  long  as  the  5-inch  line.  How 
much  longer  is  it  than  the  5-inch  line  ? 

57.  Draw  a  line  -|  as  long  as  the  5-inch  line.     -|.     ^. 

58.  Mary  has  5  cents,  and  Anna  has  ^  as  much.     How 

many  cents  has  Anna  ?     John  has  ^  as  much  money  as 

Mary.      How  many  cents  has  John  ?     Kate  lias  |^  as  much 

as  Mary.     How  many  cents  has  Kate  ?     Thomas  has  |  as 

much  as  Mary.     How  many  cents  has  Thomas? 

Illustrate  with  actual  mouey  if  the  children  fail  to  think  out  this 
work. 

59.  1^  of  anything  is  how  much  more  than  the  whole  of 
it?     ^  is  how  much  more  than  the  whole?     -|?     J^? 

60.  Numbers  like  -i,  J,  ^,  that  show  parts  of  anything 
are  called  Fractions.     Write  some  other  fractions. 

61.  DraAV  a  line  3  in.  long  and  another  line  ^  longer. 
How  many  inches  in  the  long  line  ? 

62.  Show  on  the  number  table  1  of  25.  Show  -|  of  25  ; 
f  of  25  ;  I  of  25. 

63.  If  John  had  25  cents  and  James  had  ^  as  much, 
how  many  cents  would  James  have  ? 

64.  Make  story  problems  about  fifths  of  25. 

65.  How  many  fives  must  be  added  to  20  to  equal  35  ? 
45  ?     30  ? 

66.  How  many  fives  must  be  subtracted  from  60  to 
leave  45  ?     To  leave  35  ?     50  ?     40  ?     20  ?     30  ?     55  ? 


no 


FIVES 


67.  3  fives  are  how  many  more  than  13  ?     6  fives  —  3  =  ? 

68.  What  must  be  added  to  8  fives  to  erjual  43  ?  To  7 
fives  to  equal  39  ? 

69.  What  must  be  subtracted  from  1(3  to  leave  3  fives  ? 
2  fives  ?     1  five  ? 

70.  33  is  how  much  more  than  6  fives  ?     Than  5  fives  ? 

71.  3  fives  +  4  =  ?     4  fives  +  2  =  ?     8  fives  +  3  =  ? 

72.  How  much  does  49  lack  of  being  equal  to  10  fives  ? 
47  is  how  many  more  than  9  fives  ? 

73.  Can  you  bring  in  (or  name )  a  flower  that  has  five 
petals  ?  How  many  petals  would  7  such  flowers  have  ? 
9  such  flowers  ? 

74.  How  many  school  days  in  3  weeks  ?  5  weeks  ?  7 
weeks  ?     11  weeks  ? 

75.  How  many  cents  =  10  nickels  and  3  cents  ?  8  nick- 
els and  4  cents  ?     4  nickels  and  5  cents  ? 

76.  (3  times  5  pk.  =  liow  many  bu.  and  pk.  ? 

77.  How  manv  tens  =  4  fives  ?     6  fives  ?     10  fives  ? 


Blackboard  Exercise 


Divide  quickly  each  num- 
ber on  the  edge  of  the  ti'i- 
angle  by  5. 


F1VE8  111 

78.  The  number  that  sliows  how  many  times  one  num- 
ber is  contained  in  another  is  called  a  Quotient.  What  is 
the  quotient  of  25  divided  by  5  ?     16  divided  by  2  ? 

79.  Give  quotients  of  50  -r-  10  ;  30  -f-  5  ;  24  ^  2 ;  55  -^  5. 

80.  -ig5-  =  ?  (This  is  read  "  15  divided  by  5  "  or  "  15 
fifths.") 

on        5^0_9       10.  _y       _2  2._?       40  —  9       60_?        70_? 
°-^'      10  ~"   •        10  ~  •         2     ~  •  5  •  5    ~  •        10  ~  • 

10x4        >,  Show  the  process  of  cancellation  and  let 

v>  v^  -)         ■  the  pupils  prove  by  trial  with  small  num- 

bers that  the  same  result  is  obtained  as  by 
dividing  the  i^roduct  of  the  numbers  above  the  line  by  the  product  of 
those  below.  Do  not  attempt  to  give  the  underlying  principles,  as 
the  power  to  perceive  them  usually  comes  at  a  much  later  stage  of  the 
child's  psychological  development. 

Q3_  10x20x11  _,    Q5^    33x5x8^,    ^^     0x5x11^, 
2x5x55       '         *   11x2x15     "         '2x44x25 

84.    222ll^ili^=v        86.    Ji^^^y  88.         ^QX^       =? 

2x11x20        2x3x2         2x10x11 

The  division  of  one  indicated  product  by  another  by  cancellation 
may  be  made  an  interesting  class  exercise,  and  it  is  very  useful  in 
helping  children  to  Ijecome  expert  in  recognizing  ratios.  As  new 
numbers  and  their  multiples  are  learned  give  class  exercises  in  this 
work,  combining  the  new  numbers  with  those  previously  learned. 

89.  Show  on  the  number  table 

I  of  35  ;  f  of  35  ;  |  of  35  ;  f  of  35  ;   4  of  35. 

90.  If  Thomas  had  35  cents  and  William  had  ^  as  many, 
how  many  did  AVilliam  have? 

91.  Make  story  problems  about  sevenths  of  35. 

92.  Kind  the  6th  multiple  of  5  and  add  7  to  it. 

93.  Add  8  to  the  4th  multiple  of  5.  To  tlie  7th.  To 
tlie  9th  ?     To  the  11th  ? 

94.  Take  6  from  the  12th  multiple  of  5.  From  the  9th. 
From  the  7th?     From  the  4th  ?     From  the  11th  ? 


112  FIVES 

95.  Add  2  tens  to  the  6th  multiple  of  5.     To  the  9th. 

96.  Add  3  twos  to  the  3d  multiple  of  5.     To  the  7th. 

Let  pupils  compose  similar  questions  and  briug  them  to  the  recita- 
tion to  be  solved  by  their  classmates. 

97.  A  child  was  asked,  "  What  is  a  multiple  of  5  ?  " 
She  answered,  "  The  number  you  get  when  you  multiply  5 
by  any  number  is  a  multiple  of  5."  Was  she  right? 
Explain. 

98.  What  is  a  multiple  of  10  ?     A  multiple  of  2  ? 

99.  What  number  is  the  fourth  multiple  of  10  ?  What 
number  must  10  be  multiplied  b}^  to  give  the  fourth  mul- 
tiple of  10  ? 

100.  By  what   must  5  be  multiplied  to  give  the  tliird 
multiple  of  5  ? 

101.  By  what  must  5  be  multiplied  to  give  45  ?    Which 
multiple  of  5  is  45  ? 

102.  If  you  Avere  to  spend  5  minutes  a  da}^  playing  with 
a  kitten,  how  much  time  would  you  spend  in  a  week  ? 

103.  A  pansy  has  5  petals.     How  many  petals  do  9 
pansies  have  ? 

104.  At  5  dollars  apiece,  how  much  will  11  hats  cost  ? 
7  hats  ?     A  dozen  hats  ? 

105.  At  $5  apiece,  how  many  hats  can  be  bought  for 

$40?     160?     125? 

106.  At  5  cents  apiece,  how  many  oranges  can  be  bought 
for  30  cents  ?     45  cents  ?     20  cents  ?     b5  cents  ? 

107.  Find  sums  :  108.    Find  differences  : 

i^3.15      111.55        111.15      129.65       $69.57       158.58 

6.75  38.57  67.25        13.17  32.85         12.95 

5.76  24.55  16.75 


FIVES  113 

109.  Add  8  thousand  2  hundred  86  to  9  thousand  3 
hundred  74. 

110.  From  5  thousand  3  hundred  24  take  2  thousand  1 
hundred  95  and  mark  the  subtrahend. 

111.  Anna  bought  some  groceries  for  her  mother.  She 
paid  ^1.15  for  tea,  if) 3. 37  for  flour,  and  til.2o  for  sugar. 
How  much  was  the  whole  bill  ? 

112.  Mr.  Williams  paid  a  doctor's  bill  of  815.75.  He 
gave  4  five-dollar  bills.  How  much  change  should  he 
receive  ? 

113.  Make  story  problems  about  buying. 

114.  Name  the  multiples  of  5  that  are  even  numbers. 
What  figure  does  each  of  the  even  multiples  of  five  end  in  ? 

115.  Name  the  multiples  of  5  that  are  odd  numbers. 
What  figure  do  they  end  in  ? 

116.  Write  the  odd  multiples  of  5  in  a  horizontal  line. 
Think  how  the  number  table  of  five  looks,  and  write  the 
multiples  of  5  that  are  even  numbers  in  a  horizontal  line 
under  the  line  you  have  just  written.  Leave  space  be- 
tween the  lines  as  in  the  number  table. 

117.  Write  in  Roman  notation  all  the  multiples  of  5  up 
to  100. 

118.  M  stands  for  1000  in  Roman  notation.  Write  in 
Arabic  notation  MC,  MCCC,  MD,  MDC,  MDCCC, 
MDCCCC,  MDCCCLIX. 

119.  Write  in  Roman  notation  1800,  1830,  1840,  1850, 
1860,  1890,  1896,  1861,  1876,  1883. 

120.  Write  in  Roman  notation  the  number  of  the  page 
on  which  you  are  reading  ;  the  number  of  the  page  on 
which  the  7th  chapter  of  this  book  begins  ;  the  number  of 
the  page  on  which  the  12th  chapter  begins. 

HOKX.   ARITH.  8 


CHAPTER   VIII 

ELEVENS 
Written  Multiplication,  Pkoduct 

How  many  units  in  each  answer  ? 
How  many  tens  ? 

How  many  units  and  how  many 
tens  in  each  answer  ? 

3.    Write  9  elevens  and  find  their 
sum. 

-j-j  4.    Find  the  sum  of  7  elevens. 

—     10  elevens. 

NUMBER  TABLE 


1.  Add  ; 

11 

11 

11 

11 

11 

2.  Add : 

11 

11 

11 

11 

11 

11 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

5i] 

66 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

114 


ELEVENS  115 

5.  Begin  at  11  and  count  by  elevens  until  you  reach 
99.  How  many  multiples  of  eleven  are  there  in  the  first 
hundred  numbers  ?     Learn  them. 

6.  Begin  at  99  and  count  backwards  by  elevens 
rapidly. 

7.  Fill  out  and  learn  the  table  beginning  "  Once  11  is 
eleven,"  and  ending  ''12  times  eleven  are  182." 

8.  AVhat  is  the  third  multiple  of  11  !  Mx  */  8th  '^  6th  ? 

Call  attention  to  the  fact  that  the  od  multiple  of  11  is  expressed 
by  two  3's,  the  5th  multiple  by  two  5's,  etc. 

9.  How  many  elevens  in  41  ?     (3(3?     77?     33?     121? 

10.  11  multiplied  by  5  =  ?     11  x  8  =  ?     11  x  4  =  ? 

11.  Add  tAvo  elevens  to  33.     55.     22.     77.     44.     QQ. 

12.  How  many  elevens  must  be  added  to  22  to  equal 
55?     44?     66?     88? 

13.  Take  2  elevens  from  77.     From  44.     88.     22.     m. 

14.  How  many  elevens  can  be  taken  from  99?  From 
132?     110? 

15.  How  many  elevens  must  be  taken  from  77  to  leave 
44?     22?     ^b'^     33?     m^! 

16.  How  many  elevens  must  be  taken  from  55  to  leave 
4  elevens  ? 

17.  Name  multiples  of  11,  and  take  elevens  from  them. 

18.  If  you  had  22  cents  and  your  mother  gave  you  11 
cents,  how  many  cents  would  you  have?  How  many 
dimes  and  cents  ? 

19.  Mary  had  44  cents,  Julia  had  11  cents  more  than 
Mary.     How  many  cents  did  Julia  have? 

20.  Eight  boys  gave  11  cents  apiet^e  toward  a  picnic. 
How  many  did  they  all  give? 


llg  ELEVENS 

21.  John  solved  11  problems  on  Monday  and  twice  as 
many  on  Tuesday.      How  many  on  both  days? 

22.  Make  problems  with  the  number  11. 

23.  Which  is  the,  greater,  57   or  5  elevens,  and  how 
much  ? 

24.  Which  is  the  greater,  6  tens  or  5  elevens,  and  how 
much  ? 

25.  Which  is  the  greater,  11  fives  or  5  elevens,  and  how 
much  ? 

26.  How  much  does  42  lack   of  being  as  great  as  4 
elevens  ? 

27.  How  much  does  58  lack  of  being  as   great  as    6 
elevens  ? 

28.  26  is  how  many  more  than  2  elevens? 

29.  91  is  how  many  less  than  9  elevens  ? 

30.  49  is  how  many  less  than  5  elevens  ? 

31.  6  elevens  — 7  =  ?     7  elevens  — 8  =  ?     4  elevens  — 9  =  ? 

For  a  class  exercise  let  the  children  choose  numbers  and  tell  how 
much  they  exceed  or  fall  short  of  multiples  of  11. 

32.  Find  the  third  multiple  of  11,  take  3  from  it,  and 
tell  how  many  tens  in  the  answer. 

33.  Find  the  5th  multiple  of  11,  take  5  from  it,  and  tell 
how  many  tens  in  the  result. 

34.  Name  the  first  multiple  of  11,  subtract  1,  and  tell 
how  many  fives  in  the  result. 

35.  Find  the  second  multiple  of  11,  subtract  2,  and  tell 
how  many  fives  in  the  result. 

36.  Add  5  to  the  4th  multiple  of  11.     Add  7  to  the 
6th  multiple  of  11. 

37.  How  many  elevens  does  it  take  to  equal  the  number 
that  is  the  3d  multiple  of  11?     The  5th  multiple  of  U? 


ELEVENS  117 

38.  Think  of   different  multiples  of  11,   and  tell  how 
many  tens  and  how  many  units  in  each. 

39.  If  a  cow  gives  11  qt.  of  milk  each  day,  how  many 
qt.  will  she  give  in  a  week  ?     In  10  days  ? 

40.  If  it  takes  11  buttons  for  a  boy's  suit,  how  many 
buttons  will  it  take  for  4  suits  ?     7  suits  ?     9  suits  ? 

41.  If  11  cents  were  given  to  each  of  5  boys,  how  many 
cents  would  all  get  ? 

42.  When  tops  are  11  cents  apiece,  how  much  will  4 
tops  cost  ?     8  tops  ?     11  tops  ? 

43.  If  33  cents  were  divided  equally  among  3  boys,  how 
many  cents  would  each  receive  ? 

44.  If  55  cents  were  divided  equally  among  5  boys,  how 
many  cents  would  each  receive  ? 

45.  Of  what  number  is  11  one  half?  11  is  ^  of  what? 
i  of  what  ?     I  of  what  ?     ^  of  what  ?     ^  of  what  ? 

46.  11  is  what  part  of  33  ?    Of  77  ?   44?    99?    55? 

47.  How  much  is  ^  of  33  ?     J  of  33  =  ? 

48.  iNIr.  Smith  had  §33  and  spent  J  of  them.  How 
many  dollars  did  he  spend  ?     How  many  had  he  left  ? 

49.  How  much  is  i  of  55  ?       |  of  55  =  ?       |  of  55  =  ? 

50.  55  cents  —  I  of  55  cents  =  ?  55  cents  —  |  of  55 
cents  =  ?     55  cents  —  -|  of  55  cents  =  ? 

51.  Make  story  problems  about  fifths  of  55. 

52.  How  much  is  1  of  77  ?  f  of  77  =  ?  f  of  77  -  ? 
f  of  77  =  ?     A  of  77  =  ?     f  of  77  =  ? 

53.  Make  story  problems  about  sevenths  of  77. 

CHART    DRILLS 

1st.  Taking  some  multiple  of  11  as  a  basis,  as  for  instance  55, 
point  to  that  and  let  the  children  give  the  numbers  that  are  equal  to 
f  of  it,  I  of  it,  4  of  it,  f  of  it,  etc. 


118 


ELEVENS 


Fig.  1 


2d.  Taking  a  multiple  of  11,  as  55,  as  a  basis,  let  the  children 
point  to  some  other  multiple,  as  3:3,  and  tell  quickly  what  part  of  55 
33  equals. 

Use  these  drills  frequently  until  the  children  can  give  the  ratios  of 
the  multiples  at  sight. 

54.  2  times  11  qt.  =  how  many  gal.  and  qt.? 

55.  4  times  11  pk.  =  liow  many  bu.? 

56.  3  times  11  ft.  =  how  many  yd.? 

57.  5  times  11  days  +  1  week  =  how  many  days? 

58.  Take  2  elevens  from  each  of  the  odd  numbers  in 

the  third  ten. 

59.  Copy  Fig.  1  by  placing  equilateral 
triangles.  How  long  would  the  perime- 
ter of  the  figure  be  if  the  side  of  eacli 
triangle  were  11  in.  long?     5  in.? 

60.  Copy  Fig.  2  by  })lacing  equilateral  tri- 
angles.     Which  is  greater,  Fig.  1  or  Fig.  2? 

Which  has  the  longer  perimeter  ? 

61.  How  long  would  the  perimeter  of  Fig.  2 
be  if  each  side   of  the  triangle  were  10  in.? 

Fig.  2        5  in.?     11  in.? 

62.  Place  7  e(pnlateral  triangles,  making  a  figure  differ- 
ent from  those  in  the  book,  and  make  problems  about  the 
perimeter  of  the  figure. 

63.  Place  equilateral  triangles  as  in  Fig.  3, 
and  find  how  long  the  perimeter  of  the  figure 
would  be  if  each  side  of  the  triangle  Avere 
11  in.  long. 

64.  Sliow  J  of  the  figure  you  have  made. 
Show  I  of  it ;    f  ;    |  ;    |  ; 


1 

8' 


1  _  5  _? 
8  8  ~  • 


2  —  ?         8_3_?         1—  how 


Fig.  3       many  eighths  ? 


ELEVENS  119 

65.  Can  you  separate  the  figure  into  4  equal  parts 
shaped  just  like  the  figure  itself,  only  smaller?  How 
many  eighths  in  each  of  those  ?     ^  =  how  many  eighths  ? 

66.  Place  8  equilateral  triangles  in  such  a  way  as  to 
make  a  figure  different  from  Fig.  3,  and  make  problems 
about  them. 

67.  Copy  Fig.  4  by  j^tlacing 
equilateral  triangles.  How  many 
triangles  in  Fig.  4?  How  many 
triangles  would  it  take  to  make  9 
such  figures  ?  7  such  figures  ?  5 
such  figures  ?     8  such  figures  ?  ^^' 

68.  How  many  such  figures  could  be  made  from  44 
equilateral  triangles  ?  From  99  equilateral  triangles  ? 
From  33  equilateral  triangles  ?  From  Q6  equilateral 
triangles  ? 

69.  Draw  a  horizontal  line  11  in.  long,  marking  the 
inches.  1  inch  is  what  part  of  it  ?  2  in.  is  what  part  of 
11  in.?  3  in.  is  what  part  of  11  in.?  5  is  what  part  of 
11  ?     7  is  what  part  of  11  ? 

70.  Draw  a  line  that  is  ^^  as  long  as  an  11-inch  line. 
Draw  a  line  that  is  ^|  as  long  as  an  11-inch  line. 

71       11 ^3_  _  _?_  11 9_  _  _?_  __9 7_  —  JL 

'■^*    11      11~11*  11       11  ~  11*  11       11~11 

Let  these  subtractions  be  shown  objectively  if  necessary. 

72.  How  many  players  in  4  football  teams  ?  In  9  foot- 
ball teams  ? 

73.  Write  in  Roman  notation  all  the  multiples  of  11 
that  are  less  than  135. 

74.  A  string  77  in.  long  can  be  cut  into  how  many 
strings  11  in.  long  ? 

75.  Make  a  drawing  of  two  rows  of  squares,  11  squares 
in  a  row,  and  tell  how  many  squares  in  it. 


120  ELEVENS 

76.  Make  3  rows  of  11  squares  each,  and  tell  how  many 
squares  there  are.  Find  the  middle  square  of  the  middle 
row  and  write  the  first  letter  of  your  name  in  it. 

77.  What  is  the  quotient  of  44  divided  by  11  ? 

78.  55  -r-  11  =  ?        11  =  ?        H  =  ?        £6  ^  ?        3  3  _  0 

11  11       •        11       •        11  ~  • 

See  note  after  Ex.  88,  p.  111. 

79.  Show  by  grouping  numbers  on  the  number  table 
how  many  twos  equal  2  elevens  ;  how  many  fives  equal 
5  elevens. 

80.  Write  the  multiples  of  11  in  the  same  position  that 
they  liave  in  the  number  table. 

81.  Multiply     11        11        11         11         11         11         11 

by^_8_6_5J7_9_8 

Lead  the  children  to  see  that  they  can  get  tlie  same  result  by  mul- 
tiplying the  units  and  then  the  tens  as  by  combining  numbers  on  the 
number  table,  and  in  an  easier  way. 

82.  Multiply     111      112      113      211      131      142      111 

by     _3     _4     _2     _3      __3      _2  8 

83.  Wlien  one  number  is  nuiltiplied  by  another,  the 
result  is  called  a  Product.  What  is  the  product  of  111 
and  4  ? 

84.  Find  the  product  of  121  and  3;  221  and  4;  122 
and  3  ;   512  and  3  ;  512  and  4. 

85.  If  one  side  of  a  square  were  321  ft.  long,  how  long 
would  the  perimeter  of  the  square  be  ? 

86.  If  each  side  of  an  equilateral  triangle  were  133  ft. 
long,  what  would  be  the  length  of  the  perimeter  of  tlie 
triangle  ? 

87.  Multiply      35  It  is  left  for  the  teacher  to  show  that  the 

l)v        4     ^'^'^  teufi  obtained  by  multiplying  the  5  units 

by  4  must  be  added  to  the  12  tens  obtained 

140      l)y  multiplying  the  3  tens  by  4. 


ELEVENS  121 

88.  Find  products:  125   152   251   215   255   515 

4    8    7    6    5  9 

89.  Multiply  555  by  each  of  the  numbers  that  are 
greater  than  1  and  less  than  10. 

90.  Multiply  2222  by  each  of  the  numliers  greater 
than  1  and  less  than  10. 

91.  Add  413  to  itself  and  see  if  the  sum  is  826. 

92.  Add  312  to  itself  and  312  to  their  sum,  and  see  if 
the  answer  is  936. 

93.  Add  121  to  121,  and  keep  on  adding  121  until  you 
have  484. 

94.  Add  211  to  211,  and  keep  adding  211  until  you 
get  1055.     How  many  211's  does  it  take  to  make  1055? 

95.  Can  you  hnd  a  better  way  of  finding  the  sum  of 
five  211's  than  by  adding  them  ?    If  not,  ask  your  teacher. 

96.  Find  the  sum  of  three  125's.  Four  215's.  Six 
512's.     Five  511's. 

97.  Write  the  5th  multiple  of  11,  under  it  the  Tth  mul- 
tiple of  5,  under  it  the  10th  multiple  of  2,  under  that  the 
6th  multiple  of  10,  and  add. 

98.  Write  the  first  odd  number  in  the  4th  ten,  under 
that  the  last  even  number  in  the  4th  ten,  under  that  the 
first  odd  number  after  30,  under  that  the  first  odd  number 
after  37,  and  find  their  sum. 

99.  Thomas  paid  #8.25  for  a  suit  of  clothes,  i^l.25  for 
some  handkerchiefs,  -f  .37  for  a  necktie,  and  i  .25  for  some 
collars.     How  much  was  the  whole  bill  ? 

100.  He  gave  the  clerk  a  ten-dollar  bill  and  a  five-dollar 
bill.      How  much  change  should  he  get  ? 

101.  How  much  will  7  horses  cost  at  #125  apiece? 

102.  Tell  what   these   words   mean  :    Sum^  Difference^ 
Product^  Quotle7it. 


CHAPTER   IX 
NIISTES 

Multiplier,  Square  Yard,  Square  of  a  Number,  Divisor 


NUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

n 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

61 

74 

84 

94 

5 

15 

25 

35 

45 

^^ 

Q^ 

75 

85 

95 

6 

16 

26 

36 

40 

m 

m 

76 

86 

96 

T 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1.  Begin  with  9,  and  learn  to  count  quickly  by  nines 
to  99. 

2.  Learn  the   multiples  of   9   that  are   less  than   100. 
How  many  are  there  ?     What  is  the  next  multiple  of  9  ? 

3.  Begin  with  108  and  count  backwards  by  nines  to  0. 

4.  Fill  out  and  learn  the  table  of  nines  ending  with 
12  times  9  =  108. 

122 


NINES 


123 


BLACKBOARD    EXERCISE 

Lead  the  children  to  see  that  as  9  falls 
1  short  of  10,  2  nines  fall  2  short  of  2  tens, 
;>  nines  fall  3  short  of  3  tens,  and  so  on. 
Will  not  some  child  discover  that  in  each 
of  the  first  ten  multiples  of  9  the  sum  of  the 
digits  is  9  ? 


5.  What  is  tlie  4tli  multiple  of  9  ?    6th  ?     8th  ?    9th  ? 

6.  How  maiiv  nines  in  63  ?    81  ?    45  ?    99  ?    54  ?    36  ? 

7.  72  +  9  =  ?     54  +  2  nines  =  ?     81  +  2  nines  =  ? 

8.  63  -  2  nines  =  ?     36  -  2  nines  =  ?     81-2  nines  =  ? 

9.  \\)  what  nuniljer  nmst  9  be  multiplied  to  give  the 
produetSl?    36?    63?    108?    54?    72?    99?    45?    18? 

10.  Multiply      29      119      119      219      259      295      295 

by        3         3         4         5         6         7  8 

11.  A  number  that  is  used  to  multiply  another  number 
is  called  a  Multiplier.     Name  the  multipliers  in  Ex.  10. 

12.  Use  4  as  a  multiplier  of  99. 

13.  Use  5  as  a  multiplier  of  999. 

14.  Use  6  as  a  multiplier  of  9999. 

15.  How  many  nines  must  be  added  io  27  to  equal  45? 
36?     54?     72?     63? 

16.  How  many  nines  must  be  taken  from  72  to  leave  54  ? 
63  ?     45  ?     27  ? 

17.  How  many  nines  must  be  taken  from  54  to  leave  5 
nines  ?     3  nines  ? 

18.  How  many  nines  must  be  taken  from  45  to  leave  2 
nines  ?     4  nines? 


124  NINES 

19.  Write  in  Roman  notation  all  the  multiples  of  9  that 
are  less  than  109. 

20.  Which  multiple  of  9  is  99  ?     36  ?     63  ?     27  ?     72  ? 

21.  Draw  3  rows  of  squares,  9  squares  in  a  row.  Keep 
on  adding  rows  of  squares  until  you  have  as  many  rows 
as  there  are  squares  in  a  row.     How  many  squares  in  all  ? 

22.  If  a  rectangle  is  just  as  long  as  it  is  wide,  it  is  a 
perfect  square.  Is  your  drawing  a  j)erfect  square  ?  Find 
the  middle  square  and  make  the  sign  of  multiplication  in  it. 

23.  Place  4  squares  so  as  to  make  a  perfect  square. 
How  long  is  one  side? 

24.  Place  9  squares  so  as  to  make  a  perfect  square. 
How  long  is  the  perimeter? 

25.  Place  4  rows  of  squares,  4  squares  in  each  row. 
How  many  squares  in  all?     Is  the  figure  a  perfect  square? 

26.  If  you  place  4  rows  of  squares,  5  squares  in  a  row, 
will  the  figure  be  a  perfect  square  ?  If  not,  what  can  be 
added  to  it  to  make  it  a  perfect  square?  What  can  be 
subtracted  from  it  to  leave  a  perfect  square? 

27.  How  many  square  inches  in  a  square  whose  sides 
are  each  5  in.  ?     2  in.  ?     10  in.  ?     9  in.  ?     11  in.  ? 

28.  John  may  draw  on  the  floor  a  square,  a  side  of 
which  is  3  ft.  long,  and  mark  it  off  into  square  feet. 
How  many  square  ft.  in  it? 

29.  A  square  measure  which  is  3  ft.  long  and  3  ft. 
wide  is  called  a  Square  Yard.  How  many  square  feet 
make  a  square  yard? 

30.  How  many  square  ft.  in  4  sq.  yd.?  In  7  sq.  yd.? 
3  sq.  yd.  ?  11  sq.  yd.  ?  5  sq.  yd.  ?  8  sq.  yd.  ?  2  sq.  yd.  ? 
9  sq.  yd.  ?     6  sq.  yd.  ?     12  sq.  yd.  ? 


NINES  X25 

31.  How  many  square  yards  in  36  sq.  ft.  ?     In  54  sq. 

ft.  ?     99  sq.  ft.  ?     45  sq.  ft.  ?     108  sq.  ft.  ?     27  sq.  ft.  ? 

32.  1  square  foot  equals  what  part  of  a  square  yard? 

33.  What  fraction  of  a  sq.  yd.  is  2  sq.  ft.  ?  3  sq.  ft.  ? 
5  sq.  ft.  ?     8  sq.  ft.  ?     6  sq.  ft.  ? 

34.  Show  ^-  of  the  square  yard  drawn  on  the  floor. 
How  many  ninths  does  it  equal? 

35.  Show  f  of  the  square  yard  and  tell  how  many 
ninths  it  equals. 

36.  Multiply  9  by  itself. 

37.  When  a  number  is  multiplied  by  itself,  the  result  is 
called  the  Square  of  that  number.  What  is  the  square  of 
9?     2?     10?     5? 

38.  Add  the  square  of  2  to  the  7th  multiple  of  9. 

Exercises  like  the  following  are  useful :  "Take  the  square  of  5,  add 
5,  take  |,  add  5,  take  ^,  square,  add  the  square  of  2,  subtract  10,  add  1, 
divide  by  10,  add  the  second  multiple  of  5,  take  ^,  add  the  square  of 
3,"  etc. 

39.  Copy  Fig.  1  by  placing  equilateral 
triangles.  If  a  side  of  each  of  the  tri- 
angles you  use  were  9  in.  long,  how  long 
would  the  perimeter  of  your  figure  be? 

40.  Find  the  length  of  the  perimeter 
of  the  figure  when  each  side  of  the  small  "fig~T 
triangles  is  5  in.     10  in. 

41.  How  many  small  triangles  in  the  large  triangle  that 
you  have  made? 

42.  How  many  triangles  in  6  such  figures?  In  8  such 
figures?  11  such  figures?  4  such  figures?  7  such  fig- 
ures?    9  such  figures?     10  such  figures?     5  such  figures? 


126 


NINES 


43.  How  many  figures  like  that  you  have  made  could 
be  made  from  18  small  triangles?  From  72  small  trian- 
gles?    54?     99?     63?     36?     81?     45?     27? 

44.  Show  -^  of  the  figure  you  have  made.     Show  J  of 


it.     Show  -^  of  it. 


Show  I  of  it. 


45. 


9. 
9 


1  —  1 
9  ~"   9' 


8.  _   3 
9  9 


6.  _   3.  _  ? 
9  9  ~   • 


5  _? 
9         • 


Fig.  2 


46.  Separate  your  figure  into  thirds 
as  in  Fig.  2.  How  many  ninths  in 
each  third  ? 

47.  -1  =  how  many  ninths  ?  |  =  how 
many  nintlis  ? 

48.  1  is  what  part  of  9  ?  2  is  what 
part  of  9  ?     4  is  what  part  of  9  ?     5  is 

what  part  of   9  ?     8  is  wliat  part  of  9  ?     9  is  how  many 
ninths  of  9  ?     3  is  Avhat  part  of  9?     6  is  what  part  of  9? 

49.  2  times  9  ft.  =  liow  many  yd.? 

50.  3  times  9  sq.  ft.  =  how  many  sq.  yd.? 

Children  sometimes  fail  to  distinguish  linear  yards  and  square 
yards.  Whenever  their  imagery  of  these  becomes  confused  or  indefi- 
nite, refer  them  to  the  actual  figures  drawn  on  the  floor. 

51.  4  times  9  sq.  ft.  =  how  many  sq.  yd.? 

52.  2  times  9  ph.  +  6  pk.  =  how  many  bu.? 

53.  3  times  9  qt.  —  7  qt.  =  how  many  gal.? 

54.  Write  all  the  multiples  of  9  that  are  odd  numbers 
less  than  100. 

55.  Write  all  the  even  multiples  of  9  that  are  less  than 
108. 

56.  Write  the  multiples  of  nine  from  9  to  81  in  a  slant- 
ing line  as  they  are  in  the  number  table. 

57.  Add  5  to  tlie  4tli  multiple  of  9. 

58.  Subtract  the  square  of  2  from  tlie  2d  multiple  of  9. 


NINES  127 

59.  Subtract  5  from  the  6tli  multiple  of  9. 

60.  Subtract  3  twos  from  the  3d  multiple  of  9. 

61.  Subtract  11  from  the  6th  multiple  of  9. 

62.  Subtract  2  elevens  from  the  5th  multiple  of  9. 
Let  class  prepare  similar  questions. 

63.  If  there  are  9  desks  in  each  row  and  6  rows  in  a 
schoolroom,  how  many  cliildren  can  have  desks  of  their 
own  ? 

64.  8  children  give  9  cents  each  to  a  Children's  Aid 
Society.     How  manj^  are  given  by  all  ? 

65.  11  children  give  '1.09  apiece  for  a  trip  to  the  country. 
How  much  do  they  all  give  ? 

66.  The  fare  to  Chicago  from  a  certain  city  in  Wiscon- 
sin is  8  9.  How  much  will  it  cost  8  persons  to  make  the 
trip  ? 

67.  A  round  trip  ticket  to  Chicago  from  a  town  in  Wis- 
consin costs  $  9.  How  much  will  it  cost  for  7  persons  to 
go  to  Chicago  and  back  ? 

68.  If  a  grown  person's  ticket  costs  twice  as  much  as  a 
child's,  how  much  will  it  cost  for  little  Mary  and  her  mother 
to  make  a  journey  to  Atlanta  when  Mary's  fare  is  §9? 

69.  If  a  boy's  suit  cost  f  9,  how  much  will  10  such  suits 
cost  ? 

70.  If  a  dressmaker  receives  $  9  for  making  a  dress,  how 
much  will  she  earn  by  making  11  such  dresses?     7?     4? 

71.  It  a  man  earns  9  dollars  in  a  week,  how  much  will 
he  earn  in  7  weeks ?     9  weeks?     12  weeks?     6  weeks  ? 

72.  If  18  cents  are  divided  equally  between  two  boys, 
how  many  cents  will  each  boy  receive  ? 

73.  If  a  ball  costs  9  cents,  how  many  balls  can  be  bought 

for  !^. 27?     1.63?     1.36?     |.81?     11.08'? 


128  NINES 

74.  What  is  the  quotient  of  45  divided  by  9?    36^9  =  ? 

See  note  after  Ex.  88,  p.  111. 

75.  Which  is  greater   and  how  much,  3  nines  or  28  ? 
34  or  4  nines  ?     58  or  6  nines  ?     55  or  7  nines  ? 

76.  8  nines  are  how  many  more  tlian  70?     How  many 
less  than  80  ? 

77.  5  nines  —  4  =  ?      7  nines  —  5  =  ?      6  nines  —  7  =  ? 

78.  How  much  does  61  lack  of  being  equal  to  7  nines  ? 

79.  How  much  do  5  nines  lack  of  being  equal  to  48  ? 

80.  How  much  do  6  nines  lack  of  being  equal  to  6  tens  ? 

81.  3  tens  —  3  nines  =  ?     5  elevens  —  5  nines  =  ? 

82.  How  many  nines  and  how  many  over  in  28  ?     38  ? 

83.  Choose  numbers  less  than  100  that  are  not  multiples 
of  9,  and  tell  hoAV  many  nines  in  them  and  how  many  over. 

84.  Turn  to  the  number  table  of  9  and  the  number  table 
of  5,  and  show  which  is  greater,  5  times  9  or  9  times  5. 

85.  Compare  9  x  11  and  11  x  9.     9  x  10  and  10  x  9. 
9  X  2  and  2  X  9. 

86.  What  is  the  product  of  90  multiplied  by  4  ?     By  7  ? 

87.  Give  tlie  product  of  9  maltiplied  by  20.     30.     40. 

88.  How  much  is  ^  of  9  ?     1  nine  and  1  of  9  ? 

89.  What  is  the  product  of  9  multiplied  by  5 J  ?    2 J  ? 

90.  Show  on  the  number  table  the  product  of  9  by  1^. 
By  31.     81      61.     41.     101      71      91      51      21 

91.  What  is   the  product  of   9  by  1|?     By  6f  ?    4|? 

2|?     8|?     lOf?     7|?     5f?     3f?     9|? 

Give  exercises  like  Ex.  90  and  91  until  pupils  are  prompt  in  that 
work.     Give  similar  exercises  on  each  number  as  it  is  taken  up. 

92.  9  is  ^-  of  wliat  number  ?     9  is  J  of  what  ?    ^  of  what  ? 
^  of  what  ?     -^  of  what  ?     ^  of  what  ?     ^  of  what  ? 


NINES  129 

93.  9  is  what  part  of  27  ?  54  ?  36  ?  99  ?  63  ?  81  ?  45  ? 

94.  What  is  J  of  27  ?  How  much  will  ^  of  a  yard  of 
ribbon  cost  at  27  cents  a  yard  ?  How  much  will  |  of  a 
yard  cost  ? 

95.  ^  of  36  =  ?  If  you  have  36  marbles,  and  lose  ^  of 
them,  how  many  marbles  will  you  lose  ?  How  many  Avili 
you  have  left  ? 

96.  Show  on  the  number  table  ^  of  45  ;  |-  of  45  ;  |  of 
45  ;   I  of  45. 

97.  Tell  what  part  of  45  is  18,  36,  9,  27. 
See  note  on  Chart  Drill  after  Ex.  53,  pp.  117,  118. 

98.  May  has  36  cents.  Ann  has  ^  as  many.  How 
many  has  Ann  ?  Louise  has  |  as  many  cents  as  May. 
How  many  cents  has  Louise  ? 

99.  John  has  |-  as  many  marbles  as  James,  who  has  45. 
How  many  marbles  has  John  ? 

100.  Make  story  problems. 

101.  Fill  out  the  following,  and  learn  to  give  the  state- 
ments in  any  order  : 

1  of  54  =  I  or  1  of  54  = 

f  or  i  of  54  =  I  or  I  of  54  = 

f  of  54  = 

102.  What  part  of  54  is  18  ?  45  ?  36  ?  27  ? 

103.  Thomas  had  J  as  much  money  as  William,  who 
liad  54  cents.     How  much  had  Thomas  ? 

Train  pupils  to  give  results  from  their  memory  of  the  ratios  of 
numbers.  When,  for  instance,  they  can  recall  the  fact  that  f  of  54  is 
45,  do  not  have  them  go  through  the  process  of  finding  i  of  54,  and 
then  f  of  it. 

104.  Make  a  table  showing  the  sevenths  of  63  from  -^  to 
1^,  like  the  table  that  shows  the  6ths  of  54. 

HORX.    ARITH.  9 


130  NINES 

Children  get  interesting  practice  for  a  short  time  from  exercises 
like  this :  "  Let  us  play  that  Mary  has  63  cents.  Louise,  how  many 
sevenths  of  Mary's  money  will  you  think  of  ? "  "I  will  think  of  f 
of  it,  or  45  cents,"  replies  Louise.  Then  other  pupils  "  think  "  and 
give  their  thoughts  promptly. 

105.  What  part  of  63  is  27?  36?  18?  54?  45? 

106.  Make  a  table  showing  the  eighths  of  72,  and  study 
it  until  you  can  tell  quickly  what  part  of  72  is  18,  63,  45, 
54,  36,  27. 

107.  Take  81  and  find  1  of  it ;  |,  J,  |,  |,  |,  |,  |. 

108.  What  part  of  81  is  18  ?  36  ?  72  ?  27  ?  63  ?  54  ?  45  ? 

109.  If  the  whole  of  anything  costs  81  cents,  how  much 
would  I  of  it  cost  at  that  rate  ?     How  much  would  |  cost  ? 

A?     5  ?    1? 
9  •      9  •      9  • 

110.  If  the  whole  of  anything  costs  81  cents,  what  part 
of  it  could  be  bought  for  9  cents  ?  18  cents  ?  36  cents  ? 
63  cents  ?     72  cents  ?     54  cents  ?     45  cents  ?     27  cents  ? 

111.  Make  story  problems  about  9ths  of  81. 

112.  What  is  iV  of  90  ?  -^^  of  90  ?  j\  of  90  ?  JL  of 
90?     -^9oOf90?     -/oOf90?     ^Vof90?     ^%oim? 

113.  What  part  of  90  is  9  ?  27  ?  18  ?  36  ?  63  ?  54  ? 
45  ?     72  ?     81  ? 

114.  Thomas  had  90  cents,  James  had  ^^  as  much  money. 
How  many  cents  had  James  ?  William  had  -^^  as  much. 
How  many  had  William  ? 

115.  If  a  yard  of  lace  costs  90  cents,  what  part  of  it 
could  be  bought  for  18  cents  ?     63  cents  ?     45  cents  ? 

116.  Make  story  problems  about  lOths  of  90. 

117.  What  is  the  quotient  when  63  is  divided  by  9  ? 

118.  A  number  that  is  used  to  divide  another  number 
is  called  a  Divisor.     Pick  out  the   divisors  :    63  -j-  9  =  ? 

33  -  11  =  ?       -^gO  =  ?       -3_6.  =  9 


NINES  131 

Divisor     9)36 

1     Quotient. 
Explain  this  as  a  new  way  of  expressing  division. 

119.  Divide  and  mark  divisors  and  quotients  : 

9)72        5)45        9)81        9)63        5)35        9)54        9)99 

120.  In  the  new  way  set  down  27  as  a  dividend  and 
some  number  that  will  exactly  divide  it  as  a  divisor  and 
write  the  quotient.  Do  the  same  with  25,  18,  44,  55,  40, 
66,  20,  50,  70,  77. 

Let  pupils  choose  other  numbers  and  their  divisors  and  find  quo- 
tients. 

121.  How  much  do  five  259's  equal  ?     Six  859's  ? 

122.  If  a  piano  costs  f  295,  what  will  7  pianos  cost  at 
the  same  price  ? 

123.  If  there  are  9  buttons  on  each  shoe,  liow  many 
buttons  are  there  on  3  pairs  of  shoes  ? 

124.  Mrs.  Smith  has  ^11.75  and  wants  to  buy  a  rock- 
ing chair  that  costs  $15.00.  How  much  more  money 
must  she  have  ? 

125.  Louisa's  mother  had  $20.  She  spent  $4.75  for 
coal,  $3.15  for  shoes,  and  $8.75  for  a  cloak.  How  much 
had  she  left  ? 

126.  Make  story  problems. 

127.  How  many  dollars  and  cents  are  9  times  $125.59? 
$212.55?     $213.39?     $991.95?     $195.59? 

128.  Write  in  Arabic  notation  MDCCCLXIX  and 
MDCCCXLIX  and  find  their  sum. 

129.  What  is  a  multiplier?  A  divisor?  The  square 
of  a  number  ?     A  square  yard  ? 

130.  Show  three  ways  of  expressing  division. 


CHAPTER  X 

THREES 

Multiplicand,  Parallel  Lines,  Trapezoid,  Ehombus, 

Eatio 

1.  Begin  with  three  and  count  quickly  by  threes  to  39. 

2.  Begin  with  39  and  count  backwards  by  threes  to  0. 

3.  Write  the  first  4  tens  in  columns,  putting  a  square 
in  the  place  of  every  third  number,  as  below. 

4.  Learn  the  missing 
multiples  of  3  and  write 
them  in  the  squares. 

5.  Write  and  learn  the 
table  of  threes  as  far  as 
"12  times  3  =  36."* 

6.  How  many  threes 
in  33?  27?  15?  12?  18? 

7.  Add  3  threes  to  21, 
27,  18,  15,  9,  30,  24,  12. 

8.  Subtract  3  threes 
from  21,  36,27,18,30,33. 

♦Playing  *' Numbers  Out,"  a  device  for  learning  the  multiplication 
table,  is  contributed  by  a  very  successful  teacher  and  v^^armly  indorsed  by 
her  pupils.  In  Numbers  Out,  the  children  stand  around  the  room,  leaving 
one  side  of  the  room  where  there  is  a  blackboard  vacant.  Beginning  at 
one  end  of  the  class,  they  number  themselves  1,  2,  etc.  In  playing 
♦*  Threes  Out,"   when  3  is  reached,  or   any   multiple  of  3,  the  child, 

132 


1 

11 

31 

3 

23 

33 

13 

23 

4 

14 

31 

5 

25 

35 

16 

26 

• 

7 

17 

37 

8 

28 

38 

19 

29 

10 

20 

40 

TIIiiEES  133 

9.    4  threes  +  2  threes  =  ?     11  threes  —  2  threes  =  ? 

10.  How  inaiiy  threes  must  be  added  to  24  to  equal  30  ? 

11.  How  many  threes  must  be  taken  from  21  to  leave 
18?     12?     9?     15? 

12.  What  number  will  be  equaled  by  adding  2  to  6 
tln-ees  ?  By  adding  1  to  9  threes  ?  By  subtracting  1 
from  10  threes  ?     By  subtracting  2  from  8  threes  ? 

Call  for  similar  questions. 

13.  Multiply      13      13      13      13      13      13      13      13 

by^        3^^_6_7_8_9 

14.  Find  products  of  23  multiplied  by  each  of  the  num- 
bers from  2  to  9. 

15.  Use  as  a  multiplier  of  53  each  of  the  numbers  from 
2  to  9. 

16.  A  number  which  is  multiplied  is  called  a  Multipli- 
cand.    Name  the  multiplicand  in  Ex.  15.     In  Ex.  13. 

17.  Use  33  as  a  multiplicand  with  each  of  the  numbers 
that  are  greater  than  one  and  less  tlian  10  as  multipliers. 

18.  Show  by  grouping  on  the  number  table  which  is 

greater,  9  x  3  or  3  x  9,    11  x  3  or  3  x  11,    5  x  3  or  3  x  5. 

Turn  to  number  tables  in  advance  and  let  pupils  show  the  equality 
of  8  X  3  and  3  x  8,  7  x  3  and  3  x  7,  etc. 

19.  Draw  on  the  boai'd  a  horizontal  line  1  foot  long  and 
show  how  many  times  a  3-incli  line  can  be  measured  off 
upon  it. 

instead  of  calling  the  number,  says  "Out,"  goes  to  the  blackboard, 
writes  his  number  large  and  bold  as  high  as  he  conveniently  can,  and 
takes  his  stand  under  it.  When  a  suthcient  number  of  children  are  out, 
the  teacher  calls  on  them  to  make  statements  about  their  numbers.  "I 
stand  for  27  or  9  threes,"  says  one.  "18  is  my  number.  It  equals 
6  threes,"  says  another.  A  child  who  in  numbering  around  names  a 
multiple  of  3,  or  who  says  "  Out "  for  any  number  that  is  not  a  multiple 
of  3,  or  who  makes  a  wrong  statement  about  his  number,  misses  the  game. 


134  THREES 

20.  How  many  times  can  a  3-inch  line  be  laid  off  upon 
a  9-incli  line  ?    Upon  a  15-inch  line  ?    Upon  a  27-inch  line  ? 

21.  A  3-inch  line  equals  what  part  of  a  12-inch  line? 
Of  a  15-inch  line  ?     Of  a  27-inch  line  ? 

22.  3  is  |-  of  what  number  ?  1  of  what  ?  yL.  of  what  ? 
J  of  what  ?     -J-  of  Avhat  ?     ^  of  wliat  ?     -^^  of  what  ? 

23.  -V_  =  9      3^6.  =  '/      24  -  3  =  ?      -1/  =  ?      21  -  3  =  ? 

24.  (3  X  8)  -  2  =  ?    (3  X  6)  ^  2  =  ?    (3  x  10)  -^  5  =  ? 
See  note  after  Ex.  88,  p.  111. 

Give  quotients  : 

25.  3)18         3)27         3)21  3)15         3)24         3)12 

26.  Write  9  as  a  divisor  of  each  of  the  multiples  of  9 
that  are  less  than  100,  and  give  quotients. 

27.  5  yards  of  ribbon  are  how  many  feet  long?  4 
yd.  ?     7  yd.  ?     9  yd.  ? 

28.  A  certain  room  is  21  feet  long.  How  many  yards 
of  carpet  must  there  be  in  each  strip  that  runs  the  whole 
length  of  the  room? 

29.  Measure  the  length  of  a  room  and  tell  how  many 
yards  of  carpet  it  would  take  for  each  strip. 

30.  John  lets  out  36  ft.  of  kite  string.  How  many  yd. 
of  strinor  are  let  out? 


'fc> 


31.  A  rug  is  12  ft.  long  and  9  ft.  wide.  How  many 
yd.  long  is  it?  How  many  yd.  wide?  Picture  it,  and 
show  how  many  yd.  of  binding  it  a\'ou1(1  take  to  go  all 
around  it.      (Draw  to  a  scale.) 

32.  Mary  has  a  flower  bed  3  yd.  long  and  2  yd.  wide. 
Tiiiidv  how  it  looks,  and  tell  liow  many  feet  of  border  it 
would  take  to  go  all  around  it. 


THREES  135 

33.  If  you  place  rows  of  squares,  each  row  containing 

three    squares,  until   the  figure  is  a  perfect  square,  how 

many  squares  will  there  be  in  it? 

Let  those  who  fail  to  image  rightly  do  the  actual  placing  or  draw- 
ing of  squares,  but  encourage  imagery  by  excusing  from  objective 
work  those  who  are  able  to  give  correct  results  without  it. 

34.  How  many  square  ft.  in  a  square  which  is  3  ft. 
long?     AVhat  do  we  call  such  a  square? 

35.  How  many  sq.  ft.  in  8  sq.  yd.  ?  4  sq.  yd.  ?  11 
sq.  yd.  ?     9  sq.  yd.  ?     7  sq.  yd.  ?     12  sq.  yd.  ? 

36.  Add  the  square  of  3,  the  square  of  9,  and  the  square 
of  10. 

37.  If  you  place  3  squares  in  a  horizontal  row  and  add 
equal  rows  of  squares  until  you  have  18  squares,  how 
many  rows  will  there  be  ? 

38.  If  24  squares  are  placed  in  the  same  way,  how 
many  rows  will  there  be  ? 

39.  What  number  of  cents  can  be  divided  into  five 
equal  parts  each  of  which  is  3  cents?  Each  of  which  is 
11  cents? 

40.  Draw  a  rectangle  having  horizontal  lines  5  inches 
long  and  vertical  lines  3  inches  long.  Divide  it  into 
square  incJies,  and  find  how  many  square  inches  there  are. 
How  many  rows  of  square  inches,  and  how  many  square 
inches  in  each  row  ? 

41.  When  lines  run  in  the  same  direction,  they  are  said 
to  be  Parallel  Lines.  Draw  two  parallel  horizontal  lines. 
Draw  three  parallel  horizontal  lines.  Three  parallel  ver- 
tical lines. 

42.  Draw  two  parallel  lines  slanting  downwards  to  the 
left.  Draw  three  parallel  lines  slanting  downwards  to  the 
right. 


136 


THREES 


43.  Show  parallel  lines  on  the  door  ;   on  the  window  ; 
on  your  desk. 

44.  Can  you  name  two  streets  or  roads  that  are  parallel  ? 

45.  Think  of  two  fences  that  are  parallel,  and  tell  where 
they  are. 

46.    How  many  lines   in  the  perimeter  of 
this  figure  ?     Which  of  the  lines  are  parallel  ? 

47.  If  a  four-sided  figure  has  only 
two  parallel  sides,  it  is  called  a  Trape- 
zoid.    Draw  a  trapezoid  like  this. 

48.  Draw  a  trapezoid  which  shall  be  in 
this  position. 


Draw  another  trapezoid  in  this  position. 

Let  pupils  draw  different  kinds  of  trapezoids  in 
different  positions,  and  show  parallel  lines. 


49.  Copy  Fig.  1.  Can  you  separate 
the  figure  into  two  equal  trapezoids  ? 
Show  ^  of  the  hgure.  ^  of  the  figure 
equals  how  many  sixths? 


Fig.  1 


50.    Copy  Fig.  2  by  placing  equilateral  triangles.     How 

many  triangles  in  Fig.  2?  How 
many  triangles  would  it  take  to 
make  7  such  figures  ?  To  make  9 
such  figures  ?  Can  you  divide  Fig. 
2  into  3  equal  trapezoids?  Hoav 
many  triangles  in  each  trapezoid  ? 


Fig.  2 


THREES 


137 


51.  Show  ^  of  Fig.  2.  Show  ^  of  it.  How  many 
ninths  in  i  of  it  ?  Show  ^  of  Fig.  2.  How  many  ninths 
in  I  of  it  ? 

52.  How  long  would  the  perimeter  of  Fig.  2  be  if  a  side 
of  each  triangle  were  3  in.?  How  long  if  each  side  were 
9  in.?     If  each  side  were  5  in.? 

53.  Copy  Fig.  3  by  placing  triangles.     How  many  tri- 
angles are  used?     How  long  would  tlie 
perimeter  of  the  figure  be  if  each  side 
of  the  triangles  were  9  in.  long?  3  in.? 

54.  Can   you   divide    Fig.   3   into    3 
equal  trapezoids  ? 

55.  Show  how  many  ninths  in  ^  of 
Fig.  3.     In  f  of  Fig.  3. 


Fig.  3 


56.  Copy  Fig.  4  by  placing  equilateral  triangles.     How 
many  triangles  does  it  take  ? 

57.  HoAv  long  would  the  perimeter  of  the 
figure  be  if  a  side  of  each  triangle  were  3  in. 
long?     5  in.  long  ?     9  in.? 

58.  Show  J  of  Fig.  4.  Show  ^V  ^^  ^^* 
Show  -^-Q  of  it.  How  many  tenths  equal  ^ 
of  it? 

59.  Separate  Fig.  4  into  5  equal  diamond - 

shaped  figures.     Each  diamond  is  what  fractional  part  of 
Fig.  4  ?     1  =  how  many  tenths  ? 

60.  A  diamond-shaped  figure  is  called  a  Rhombus.    How 
many  sides  has  a  rhombus  ?     Has  it  any  square  corners  ? 

61.  How  long  is  the  perimeter  of  a  rhombus,  each  of 
whose  sides  is  11  in.?     5  in.?    9  in.?     3  in.? 


138 


THKEES 


62.  Copy  Fig.  5  by  placing 
equilateral  triangles.  How- 
many  triangles  in  Fig.  5  ? 

Fig.  5  63.    Show  ^  of    the   figure 

you  have  made.     Show  -^q  of  it ;  ^^^,  -f^,  -^q. 

64.  Separate  your  figure  into  5  rhombuses,  i  =  how 
many  tenths  ?  f  =  how  many  tenths  ?  -|  =  how^  many 
tenths  ?     ^  =  how  many  tenths  ? 

65.  3  is  what  part  of  9  ?  27  ?    36  ?    18  ?    24  ?   33  ?    21  ? 

66.  What  is  i  of  15  ?     |  of  15  ?     |-  of  15  ?     |  of  15  ? 

67.  What  part  of  15  is  12  ?  9  ?  3  ?  6  ? 
See  note  on  Chart  Drills  after  Ex.  53,  pp.  117,  118. 

68.  If  a  yd.  of  cloth  costs  $.09,  how  much  will  -^  of  a 
yd.  cost  ?     I  of  a  yd.  ? 

69.  When  nuts  are  |  .12  a  pound,  what  part  of  a  pound 
can  be  bought  for  1 .03  ?  f  .09  ?  |  .06  ? 

70.  If  a  yd.  of  ribbon  costs  f  .15,  how  much  Avill  |  of  a 
yd.  cost?     I  of  a  yd.?     fyd.?     f  yd.?     |  yd.?     1 3-d.? 

71.  If  15  yd.  of  cloth  cost  a  certain  sum  of  money, 
what  part  of  the  money  will  3  yd.  cost  ?  What  part  will 
9  yd.  cost?     6  yd.?     12  yd.  ? 

72.  How  much  is  j  of  21  ?  f  of  21  ?  |  of  21  ?  f  of 
21?     I  of  21?     fof21? 

73.  What  part  of  21  is  3  ?     9  ?     18  ?     6  ?     15  ?     12  ? 

74.  If  a  yd.  of  ribbon  costs  21  cents,  how  much  wall  | 
of  a  yd.  cost  ?     |-  of  a  yd.  ?     ^  yd.  ?     -|  yd.  ?     |  yd.  ? 

75.  Take  24  and  show  what  part  of  it  is  3,  9, 1 2,  6,  18,  21. 

76.  How  many  hours  is  it  from  9  o'clock  Monday 
morning  till  9  o'clock  Tuesday  morning  ? 


THREES  139 

77.  How  many  days  from  Monday  morning  to  Wednes- 
day morning  ?  How  many  hours  /  From  9  o'clock  to 
12  o'clock  is  what  fractional  part  of  a  day  ? 

Refer  to  clock  or  watch. 

78.  What  fractional  part  of  a  day  is  6  hours  ?  9  hours  ? 
15  hours  ?     21  liours  ? 

79.  From  6  o'clock  in  the  morning  to  6  o'clock  at  night 
equals  how  many  hours  ?     What  part  of  a  day  ? 

80.  From  11  o'clock  in  the  morning  until  2  o'clock  in 
the  afternoon  equals  how  many  hours  ?  What  part  of  a 
day  ? 

81.  From  10  o'clock  in  the  morning  till  4  in  the  after- 
noon equals  what  part  of  a  day  ? 

82.  Take  each  multiple  of  3  that  is  less  than  27  and 
show  what  part  it  is  of  27. 

83.  3  equals  what  part  of  30,  or  what  is  the  ratio  of  3 
to  30  ? 

Use  these  expressions  interchangeably. 

84.  What  is  the  ratio  of  1  to  2  ?  1  to  3  ?  2  to  3  ?  1 
to  5?     2  to  5?     4  to  5? 

85.  Give  ratio  of  3  to  9.     3  to  12.  3  to  24.     3  to  15. 

86.  What  is  the  ratio  of  9  to  27  ?  To  81  ?     36  ?     72  ? 

87.  What  number  is  ^^  of  30  ?  j^^  of  30  ?  ^V  of  -30  ? 
foOf30?     ^2_of30?     ^4_of30?     fo-ofSO?     ^s^of.SO? 

If  the  children  try  to  memorize  the  statements  of  ratios  without 
perceiving  the  relations  of  numbers,  let  them  work  out  the  ratios  by 
dividing  lines  or  grouping  numbers  on  the  number  table. 

88.  What  part  of  a  yard  is  3  inches?  6  in.?  18  in.? 
24  in.?     33  in.?     21  in.?     15  in.?     9  in.?     27  in.? 

89.  What  is  the  ratio  of  a  3-in.  line  to  a  line  a  yd.  long  ? 

90.  What  is  the  ratio  of  1  to  100  ?  6  to  100  ?  10  to  100  ? 


140  THREES 

91.  Point  out  nuiiibers  on  the  number  table  and  tell 
tlieir  ratio  to  100. 

92.  What  is  the  ratio  of  a  foot  to  a  yard  ? 

93.  Think  of  a  square  foot,  and  with  your  finger  out- 
line in  the  air  its  perimeter. 

94.  Outline  in  the  air  a  square  yard.  What  is  the 
ratio  of  a  square  foot  to  a  square  yard  ? 

95.  Show  with  your  hands  as  nearly  as  you  can  the  size 
of  a  pint  measure  ;  the  size  of  a  quart  measure.  What  is 
the  ratio  of  a  pint  to  a  quart?  In  the  same  way  show 
size  of  inch  and  foot ;  peck  and  bushel ;  quart  and  gallon, 
and  give  ratios. 

96.  How  much  is  30  multiplied  by  4?  By  6  ?  8?  3? 
9?     7? 

97.  How  much  is  3  multiplied  by  20  ?  By  80  ?  30  ? 
90?     60?     40? 

98.  How  much  is  3  multiplied  by  l^  ?  21  ?  BJ-  ?  81  ? 
71?     91?     31?     51?     41?     121?     101?     111? 

See  note  after  Ex.  91,  p.  128. 

99.  4  times  ^  =  how  many  whole  ones  and  how  many 
thirds  over? 

100.  How  much  is  5  times  ^?     8  times  -^?     9  times  J? 
10  times  -J?     12  times  J?     15  times  i?     18  times  -J? 

101.  Find  quotients  : 

3)24         3)18         9)72         5)30         9)63         9)45 

102.  When  4  threes  are  subtracted  from  14  what  is  the 
remainder  ? 

103.  How  much  is  the   remainder  when   2   threes   are 
subtracted  from  8  ? 

104.  Divide  10  by  3.     What  is  the  quotient  and  what 
the  remainder? 


J 


THREES  141 

105.  jNIark  divisor,  quotient,  and  remainder  in  the  fol- 
lowing examples  : 

Divisor     5)17 

Quotient    3,  Remainder  2. 

5)26      9)38      3)13      9)19      9)78      5)39      3)19      9)65 

106.  When  a  multiple  of  3  is  divided  by  3,  is  tliere  ever 
a  remainder  ?     Explain. 

107.  Write  all  the  numbers  between  21  and  30  that  are 
not  multiples  of  3,  divide  each  of  them  by  3,  and  mark 
quotient  and  remainder. 

108.  Write  in  Roman  notation  the  first  13  multiples 
of  9. 

109.  Write  in  Arabic  notation  MDCCCXCIX  and 
MDCCCCV,  and  find  their  difference. 

110.  Find  sums  :  111.    Find  differences  : 

19.13         18.23  129.75         $18.29         8384.78 

8.23  2.43  3.83  12.63  31.96 

2.13  6.73 

112.  Jolm  has  $10.09  and  wants  to  buy  a  bicycle  that 
costs  1 25.     How  much  more  money  must  he  get  ? 

113.  John  earns  12.07  to  add  to  his  110.09.  How 
much  does  he  still  lack  ? 

114.  Some  one  gives  him  $.27.  How  much  does  he 
still  lack  ? 

115.  He  earns  $2.18  more.  How  much  does  he  still 
lack  ? 

Let  the  children  make  problems  in  which,  as  in  the  foregoing, 
there  is  a  continued  striving  toward  some  desired  end. 

116.  AVhat  is  meant  by  the  words:  Multiplicand^  Par- 
allel Lines,  Trapezoid,  Rhombus  f 


CHAPTER   XI 

EIGHTS 

Dexominator,  Quart  and  Peck,  Short  Division,  Divi- 
dend, Perpendicular  Lines,  Area  of  Right  Tri- 
angle 


"1 

NUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

7.) 
<  'J 

83 

93 

4 

14 

24 

34 

44 

54 

64 

<4 

84 

94 

6 

15 

25 

35 

45 

55 

Gh 

75 

85 

95 

6 

16 

2(J 

36 

46 

56 

Ca) 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1.  Count  by  eights  to  96.     How  many  eights  did  yon 
count  ? 

2.  Begin  with  96  and  count  quickly  hy  eights  bach  to  0. 

3.  Write  and  learn  the  table  that  ends  with  "12  times 
8  are  96." 

142 


EIGHTS  143 


o 


4.  What  is  the  3d  multiple  of  8  ?     5th  ?     9th  ?     6th  ? 
11th?     7th?     12th?     8th?     4th?     10th? 

5.  Which  multiple  of  8  is  32  ?     48  ?     64  ?     16  ?     56  ? 

6.  (iive  quotients  : 

8)16      8}48      8)64      8)80      8)56      8)32      8)72      8)24 

7.  Add  2  eights  to  40.     To  64.     48.     56.     24.    16. 
40-2  eights  =  ?       80-2  eights  =  ?       56-2  eights  =  ? 

8.  How  many  eights  must  be  added  to  24  to  equal  40  ? 
56'^     72?     48?     80?     64?     88?     96'^ 

9.  How  many  eights  must  be  taken  from  80  to  leave 
64?     48?     32?     bi)?     24?     40?     16? 

10.  How  many  eights  must  be  added  to  32  to  equal  9 
eights  ?     7  eiglits  ?     5  eights  ?     8  eights  ?     6  eights  ? 

11.  How  many  eights  must  be  taken  from  64  to  leave 

6  eights  ?      3  eights  ?      5  eights  ?      4  eights  ?      2  eights  ? 

The  game  "Eights  Out,"  like  that  described  in  the  footnote  on 
page  132,  is  useful. 

12.  5  eights  -f-  8  =  ?    7  eights  +  6  =  ?    2  eights  +  5  =  ? 

13.  Find  quotients  and  remainders  : 

8)28       8)75         8)50         8)71         8)39         8)53       8)47 

14.  How  many  must  be  added  to  21  to  equal  3  eights  ? 

15.    may  tliink  of  a  number  and  tell  how  many 

must  be  added  to  it  to  equal eights. 

16.    may  think  of  a  number  and  tell  how  many 

must  be  subtracted  from  it  to  leave eights. 

17.    may  think  of  a  number  less  than  14,  subtract 

it  from  9  eights,  and  tell  what  is  left. 

18.  Is  the  3d  multiple  of  8  even  or  odd  ?      Can  jon 

write  a  multiple  of  8  that  is  an  odd  number  ? 

Call  attention  to  the  fact  that  the  endings  of  the  multiples  of  8 
differ  by  2  in  regular  order,  8,  6,  4,  2,  0. 


144  EIGHTS 

19.  iMark  products,  multiplicands,  and  multipliers  : 

18  18  18  18  38  38  38  38 

20.  Use  28  as  a  multiplicand  with  each  of  the  numbers 
that  are  greater  than  1  and  less  than  10  as  multipliers. 

21.  Use  5  as  a  multiplier  with  119,  218,  318,  518,  918. 

22.  Eugene  sold  3  times  as  many  papers  as  his  brother, 
who  sold  28  papers.  How  many  papers  did  Eugene  sell  ? 
How  many  more  than  his  brother  ? 

23.  Use  9  as  a  multiplier  of  181,  251,  381,  581,  881,  981. 

24.  Use  82838  as  a  multij^licand  with  each  of  the  odd 
numbers  that  are  less  than  10  and  greater  than  1. 

25.  Use  85898  as  a  multiplicand  witii  each  of  the  even 
numbers  that  are  less  than  10. 

26.  What  number  must  be  used  as  a  multiplier  of  8  to 
produce  32?     24?     56?     72?    96?    64?    88?    48?    16? 

See  note  after  Ex.  88,  p.  111. 

27.  How  many  sheets  of  paper  must  be  divided  among 
5  children  to  give  each  child  8  sheets  ?  10  sheets  ?  9  sheets  ? 

28.  If  there  are  8  cherries  in  a  bunch,  how  many  cher- 
ries are  there  in  10  bunches  ?  In  12  bunches?  In  6  bunches  ? 

29.  If  it  takes  8  eggs  for  a  cake,  how  many  cakes  can 
be  made  with  2  dozen  eggs  ?     With  48  eggs  ? 

30.  Make  problems  using  the  number  8. 

31.  What  is  the  ratio  of  8  to  16  ? 

Use  chart  drill  as  suggested  in  note  after  Ex.  .53,  pp.  117,  118. 

32.  A  boy  offers  to  trade  a  big  apple  for  16  marbles. 
How  much  of  the  apple  ought  he  to  give  for  8  marbles  ? 

33.  AVhat  is  the  ratio  of  8  to  24  ?     Of  16  to  24  ? 


EIGHTS  145 

Children  slioiild  be  led  gradually  to  see  such  facts  as  that  the  ratio 
of  the  first  multiple  of  any  number  to  its  second  multiple  is  | ;  of  the 
2d  to  the  3d,  f ;  of  the  5th  to  the  10th,  ^. 

34.  Two  boys  receive  24  cents  for  cutting  some  wood. 
The  big  boy  does  f  of  the  work.  How  much  should  he 
receive  ?     How  much  should  the  small  boy  get  ? 

35.  What  is  ^  of  32?     fof32?     fori  of  32? 

36.  What  part  of  32  cents  do  16  cents  equal  ?    24  cents  ? 

37.  Mary  has  ^  as  many  cents  as  Harriet,  who  has  32 
cents.     How  many  cents  has  Mary? 

38.  If  8  men  can  do  a  piece  of  work  in  4  days,  how  long 
will  it  take  1  man  to  do  the  same  work  ?     2  men  ? 

39.  What  is  the  ratio  of  8  to  40  ?  How  much  is  f  of  40  ? 
4  of  40?     fof40? 

40.  What  part  of  40  is  16  ?     32  ?     24  ? 

41.  How  many  cents  are  |  of  40  cents?     |  of  $.40? 

42.  John  had  40  cents  and  lost  |  of  them.  How  many 
cents  did  he  lose,  and  how  many  had  he  left  ? 

43.  A  C  D  E  F  B.  The  line  AB  repre- 
sents a  distance  of  40  miles,  divided  into  5  equal  parts. 
How  far  is  it  from  A  to  (7?  ^  to  .E^?  J.  to  i)  ?  A  to  F? 
CtoE^i    BtoB?    FtoB? 

44.  What  is  the  ratio  to  the  whole  distance  of  the  dis- 
tance from  Ato  0?  AtoF?  FtoB?  BtoB?  O  to  F? 
CtoF?    AtoB?     FtoB? 

45.  Fill  out  and  learn  the  following  : 

1  of  48  =  The  ratio  of    8  to  48  is 

•|  or  |-  of  48  =  The  ratio  of  16  to  48  is 


I  or  1  of  48  =  The  ratio  of  24  to  48  is 

I  or  f  of  48  =  The  ratio  of  32  to  48  is 

f  of  48  -  The  ratio  of  40  to  48  is 

HORN.    ARITH.  10 


146  EIGHTS 

46.  Tell  quickly  what  is  the  ratio  to  48  of  each  of  the 
multiples  of  8  less  than  48. 

47.  Three  boys  caught  48  fish.  John  caught  ^  of  them, 
James  -J  of  them,  and  Henry  ^  of  them.  How  many  fish 
did  each  boy  catch? 

48.  Two  men  bought  48  bu.  of  apples,  one  man  paying 
for  J  of  them,  and  the  other  man  for  the  rest.  How  many 
bu.  ought  each  man  to  receive? 

49.  If  the  apples  cost  $  15,  how  much  should  each  man 
pay? 

50.  Make  a  table  showing  sevenths  of  56  from  y  to  |^. 

51.  Learn  to  give  quickly  the  numbers  whose  ratio  to 

.^6  I'c   5     3     6     4     2 

OU   lb   y,    ■^,    f,  y,   y. 

52.  If  56  marbles  were  divided  equally  among  7  chil- 
dren, how  many  marbles  would  2  children  receive?  How 
many  would  4  children  receive  ?     6  children  ? 

53.  Albert  missed  8  words  in  spelling  56  words.  What 
fractional  part  of  the  words  were  spelled  wrong  ?     Right  ? 

54.  John  had  56  cents  and  spent  16  cents.  How  many 
sevenths  of  his  money  did  he  spend  ?  How  many  sevenths 
did  he  keep  ?     How  many  cents  ? 

55.  Make  story  problems  about  sevenths  of  56. 

56.  Draw  on  the  board  an  8-inch  square  and  divide  it 
into  inch-squares?  How  many  rows  of  squares?  How 
many  squares  in  each  row? 

57.  How  many  squares  in  ^  of  the  figure?  In  |  of  it? 
In  f  of  it?     In  I  of  it  ?     In  i  of  it  ?     In  i  of  it? 

58.  What  part  of  the  whole  figure  are  8  squares  ?  24 
squares  ?  56  squares  ?  16  squares  ?  32  squares  ?  48 
squares?    40  squares?    1  square?    7  squares?    13  squares? 


EIGHTS  14 


n 


59.  Make  a  list  of  the  numbers  that  are  ^,  |,  |,  |,  ^,  |, 
i  I  of  72. 

60.  Learn  to  give  quickly  the  ratio  to  72,  of  24,  8,  48, 
64,  32,  ,%,  16,  40. 

61.  If  72  men  do  a  piece  of  work  in  a  day,  how  much 
of  it  is  done  by  8  men  ?  32  men  ?  40  men  ?  64  men  ? 
16  men  ? 

62.  8  has  the  ratio  ^  to  what  number?  Find  -^^  of 
that  number       -"^   -3_   _4_   __8_   _5_  _S_   _6_ 

63.  Give  quickly  the  ratio  to  80  of  each  of  the  multi- 
ples of  8  that  are  less  than  80. 

64.  If  a  yd.  of  cloth  costs  f  .80,  how  much  of  it  can  be 
bought  for  1.08?     1.16?     ^.40?     1.24?    $M?    i.64? 

65.  Make  story  problems. 

66.  In  fractions  the  number  that  is  written  below  the 
line  is  called  the  Denominator. 

Name  the  denominator  of  |-,  ^,  ^^~. 

67.  Write  and  read  a  fraction  Avith  8  as  the  denomi- 
nator and  7  for  the  number  above  the  line. 

68.  AVrite  and  read  a  fraction  with  8  as  the  denomi- 
nator and  some  odd  number  for  the  other  number. 

69.  Write  and  read  a  fraction  with  9  as  the  denomi- 
nator and  some  even  number  for  the  other  number. 

70.  Write  several  fractions  with  8  for  the  denominator 
and  some  other  multiple  of  8  for  the  number  above  the 
line.     Tell  what  each  equals. 

Will  not  the  children  see  that  a  fraction  is  a  form  of  division  ? 

71.  Write  several  fractions  with  9  for  the  denominator 
and  a  multiple  of  9  for  the  other  number,  and  tell  what 
each  fraction  equals. 


148  EIGHTS 

72.  What  is  the  value  of  the  fraction  that  has  6  for  its 
denominator  and  3  for  the  number  above  the  line  ? 

73.  What  is  the  product  of  80  multiplied  by  6?     8? 
11?     9?     7? 

74.  What  is  the  product  of  8  multiplied  by  30  ?  70  ? 
50?     90?     60?     40?     80? 

75.  What  is  the  product  of  80  multiplied  by  30  ?  40  ? 
90?     60?     70? 

76.  How  much  is  6  eights  and  |^  of  8  ?  7  eights  and 
1  of  8  ?     9  eights  and  |  of  8  ? 

77.  Louise  has  8  cents,  and  Mary  has  2|  times  as  many. 
How  many  cents  has  Mary  ? 

78.  Make  story  problems. 

79.  How  many  quarts  make  a  peck  ? 
Use  actual  measurements. 

80.  How  many  qt.  equal  3  pk.  ?  5  pk.  ?  9  pk.  ?  10  pk.  ? 

81.  Fill  out  and  learn  the  table  of  Dry  Measure. 

pints  (pt.)  =  1  quart  (qt.). 

quarts  =  1  peck  (pk.). 

pecks  =  1  bushel  (bu.). 

82.  How  many  quarts  in  3  pk.  -H  7  qt.  ?    4  pk.  +  5  qt.  ? 

83.  What  is  the  ratio  of  a  qt.  to  a  pk.  ?    3  qt.  to  a  pk.  ? 

84.  How  many  pk.  and  qt.  in  18  qt.  ?    27  qt.  ?    33  qt.  ? 

85.  Find  quotient  :     3)36 

Show  the  process  of  dividing  tens  and  units  separately.  Lead  the 
children  to  see  that  they  get  the  same  result  by  this  process  as  by 
grouping  numbers. 

86.  Find  quotients : 

5^55     2)28     9)99     8)88     2)242     2)264     3)336     2)226 


EIGHTS  149 

87.  Mary  is  making  badges  3  inches  long.  How  many 
can  she  make  out  of  a  piece  of  ribbon  63  inches  long  ? 

88.  Among  how  many  children  can  88  cherries  be 
divided,  giving  each  child  4  cherries  ? 

89.  Find  quotients  and  remainders  : 

3)964      2)245      3)865      2)4843      9)93      9)185      9)276 

90.  Use  8  as  a  divisor  of  489,  645,  568,  168,  327,  720. 

91.  A  number  which  is  divided  by  another  number  is 
called  a  Dividend.     Name  the  dividends  in  Ex.  89. 

92.  With  5  as  divisor  use  as  dividends  105,  255,  458. 

93.  AYith  9  as  divisor  use  as  dividends  279,  364,  723. 

94.  With  8  as  divisor  use  as  dividends  643,  167,  489. 

95.  If  8  squares  are  placed  in  a  row,  how  many  rows 
must  there  be  to  use  40  squares  ?     56  squares  ? 

96.  If  8  squares  are  placed  in  a  row,  how  many  rows 
must  there  be  to  make  the  figure  a  perfect  square  ?  How 
many  squares  in  the  figure  ?     What  is  the  square  of  8  ? 

97.  A  triangle  that  has  a  square  corner  is  called  a  Right 
Triangle.     Draw  a  right  triangle. 

98.  Make  right  triangles  by  bisecting  an  inch-square. 
The  lines  that  meet  to  form  the  square  corner  are  perpen- 
dicular to  each  other.  How  long  is  each  of  the  perpen- 
dicular sides  of  the  triangles  you  have  made  ? 

99.  Place  8  triangles  as  in  Fig.  1.  If  you 
take  away  the  four  outside  triangles,  what 
kind  of  a  figure  will  be  left  ? 

100.  What  is  the  ratio  of  the  figure  that 
is  left  to  the  figure  as  it  was  at  first  ? 

101.  To  how  many  square  in.  is  your  copy  ^^' 
of  Fig.  1  equal. 


150 


EIGHTS 


102.  Show  by  Fig.  1  how  many  eighths  equal  ^ ;  how 
many  eighths  equal  ^. 

103.  In  the  fraction  -^,  which  number  is  the  denomi- 
nator ? 

104.  Copy  Fig.  1  by  drawing.     Make  each  of  the  per- 
pendicular sides  of  the  triangles  one  inch  long. 

105.  Copy  Fig.  2  by  placing  triangles. 
How  many  such  figures  could  you  make 
with  24  triangles  ?    72  ?    48  ?    80  ?    96  ? 


106.  Copy  Fig.  2  by  drawing,  making 
the  perpendicular  sides  of  each  triangle 
1  inch  long.  How  many  square  inches 
in  your  copy  ? 


Fig.  2 


107.  What  is  tlie  ratio  of  one  of  the  triangles  to  the 
whole  figure  ?  Of  three  triangles  to  the  whole  ?  Of  5 
triangles  to  the  wliole  ? 

108.  How  many  sq.  in.  in  a  rectangle  8  in.  long  and  7 
in.  wide  ?  8  in.  long  and  5  in.  wide  ?  8  in.  long  and  8 
in.  wide  ?  What  is  a  rectangle  called  that  is  as  long  as  it 
is  wide  ? 

109.  Draw  two  square  inches,  divide  them  into  halves, 
and  letter  them  as  in  Figs.  3  and  4.  Which  is  greater, 
the  triangle  ABD,  or  the  rectangle  EFOD  ? 


Fig.  3 


Fig.  4 


EIGHTS 


151 


Fig.  5 


Besides  bringing  out  the  fact  that 
they  are  each  one  half  of  a  square  inch, 
let  the  equality  of  the  figures  be  shown 
by  cutting  off  the  upper  part  of  the  tri- 
angle and  fitting  it  to  the  lower  part  to 
form  a  rectangle,  as  in  these  figures. 

110.  Copy  Fig.  5  by  drawing,  making 
the  horizontal  lines  2  inches  long  and  the    ^ 
vertical  lines  1  inch  long. 

111.  How  many  square  inches  in  the  tri- 
angle 7li>C?     In  ABO? 

Let  the  children  prove  their  answers  by  cutting  and  fitting  surfaces. 

112.  How  many  sq.  in.  in  a  right  triangle  8  in.  long  and 
3  in.  wide  ?     8  in.  long  and  7  in.  wide  ? 

113.  How  many  sq.  in.  in  a  triangle,  one  of  whose  per- 
pendicular sides  is  9  in.  long  and  the  other  7  in.? 

114.  What  is  the  area  of  a  right  triangle  9  in.  long  and 
8  in.  wide  ? 

115.  What  is  the  area  of  a  right  triangle  9  in.  long  and 
6  in.  wide? 

116.  When  the  length  of  the  perpendicular  sides  of  a 
right  triangle  is  given,  how  is  the  area  of  the  triangle 
found  ? 

117.  Make  a  trapezoid  by  placing  5  equilateral  triangles. 

118.  Copy  Fig.  6  by  placing  equi- 
lateral triangles.  How  long  would  the 
perimeter  of  Fig.  6  be  if  each  side  of 
the  triangles  w^ere  8  in.?  If  each  side 
were  5  in.?     11  in.?     9  in.? 

119.  Can  you  take  away  3  rhombuses 
from  Fig.  6  and  leave  a  trapezoid? 

120.  What  is  the  ratio  of  each  rhombus  to  the  whole 
figure  ? 


Fig.  6 


152  EIGHTS 

121.    What  is  the  ratio  of  the  trapezoid  to  the  whole 

figure  ? 

122.  Copy   Fig.    7    by   placing   equi- 

ateral  triangles.  Can  you  take  away  3 
trapezoids  from  the  figure  and  leave  one 
triangle  ? 

123.  One  trapezoid  has  what  ratio  to 
the  whole  figure  ?  Two  trapezoids  have 
what  ratio  to  the  whole  figure  ? 

124.  How  long  would  each  side  of  the  triangles  be  if 
the  perimeter  of  Fig.  7  were  12  in.?     60  in.?     96  in.? 

125.  Find  quotients  : 

8)1688    8)3288    8)4880    8)6488    2)f| 

126.  Divide:     2)356 

178 

Show  this  process  of  short  division. 

127.  Divide  : 

3)726         3)654        3)9381         3)427        5)115        5)275 

5)385      5)4355  9)828         9)738       9)1269      9)1648 

128.  Mr.HoAve  divided  $2268  among  4  grandchildren. 
How  much  did  each  receive  ? 

129.  How  many  weeks  are  there  in  1323  days  ? 

130.  What  number  multiplied  by  8  will  give  968  ? 

131.  Divide  187  by  each  of  the  numbers  2,  3,  5,  8,  9. 

132.  Divide  437  by  2,  3,  5,  8,  9. 

133.  Divide  493  by  2,  3,  5,  8,  9. 

134.  Divide  the  first  odd  number  after  209  by  2,  3,  5,  8. 

135.  At  8  cents  a  qt.,  how  much  will  a  pk.  of  berries 
cost  ? 


EIGHTS  153 

136.  At  8  cents  a  qt.,  how  much  do  half  a  pk.  of  berries 
cost  ? 

137.  How  many  ounces  in  |^  a  pound  ?     In  1 J  pounds  ? 

138.  At  8  cents  apiece,  how  much  will  a  dozen  pine- 
apx:)les  cost?  How  much  change  ought  you  to  get  from  a 
dollar  bill  after  paying  for  them  ? 

139.  If  you  bought  3  yd.  of  ribbon  at  8  cents  a  yd.,  and 
gave  the  clerk  a  quarter  of  a  dollar,  how  much  change 
ought  he  to  give  you  ? 

140.  If  you  bought  6  yd.  of  calico  at  8  cents  a  yd.,  and 
gave  the  clerk  half  a  dollar,  how  much  change  should 
you  get? 

141.  Anna  bought  7  yd.  of  ribbon  at  8  cents  a  yd.,  and 
gave  the  clerk  a  half  dollar  and  a  dime.  How  much 
change  was  due  ? 

142.  Write  and  add  the  square  of  8,  the  square  of  5,  and 
the  square  of  9. 

143.  A  Sunday  school  wishes  to  buy  an  organ  which 
costs  1)125.  The  treasurer  has  $75.08.  How  much  more 
is  needed  ? 

144.  114.23  more  were  paid  in.  How  much  was  still 
needed? 

145.  After  128.13  more  were  paid  in,  how  much  was 
needed  ? 

146.  Mr.  Brown  gave  |10  toward  the  organ.  How 
much  more  was  raised  than  the  organ  cost  ? 

147.  If  the  rent  of  a  house  is  122.50  a  month,  how 
much  will  the  rent  for  the  summer  months  be  ? 

148.  Write  in  Arabic  notation,  and  divide  by  8,  CXV, 
XCV,  CLXII,  CCCCLIX,  MDCCCXC. 

149.  Explain  the  words :  Denominator^  Dividend,  Divisor, 
Right  Triangle,  Perpendicular  Lines. 


CHAPTER   XII 


FOURS 


'     Numerator,   Square  Prism,  Partial  Products,  Ton 


1.  Begin  with  4,  and  count  by  fours  to  48. 

2.  Count  by  fours  from  48  until  nothing  is  left. 

3.  Write  the  first  five  tens  in  vertical  columns,  putting 
a  square  in  the  place  of  every  fourth  number. 

1  11  21  31  41 


33 


D 


43 


3 


D 


13 


14 


33 


33 


31 


43 


n 


6 


15 


D 


35 


36 


35 


45 


46 


17 


18 


07 


D 


37 


38 


47 


9 


10 


19 


29 

30 
154 


39 


D 


49 


50 


FOURS  155 

4.  Learn  the  missing  multiples  of  4,  and  write  them 
in  the  squares. 

5.  Write  and  learn  the  table  ending  "  12  times  4  are 
48"." 

6.  Name   different  multiples  of   4,  and   tell   whether 
they  are  even  or  odd. 

7.  Name  the  3d  multiple  of  4,  the  7th,  9th,  12th,  2d. 
a  -2  0.'''  X§.'^  -1^'^  M'^  40-'^  48-''*  ^-'^  -M-'''  AA'^ 
9.    Add  2  fours  to  20,  to  36,  to  24,  to  16,  to  32. 

10.  Take  3  fours  from  48,  16,  40,  32,  44,  24,  20. 

11.  40 -2  fours  =?    48 -3  fours  =?    32 -4  fours  =? 

12.  8  fours  +  2  fours  =?     7  fours  +  5  fours  =? 

13.  7  fours  —  3  fours  =?     9  fours  —  4  fours  =? 

14.  How  many  fours  must  be  added  to  16  to  equal  28? 

15.  How  many  fours  must  be  taken  from  32  to  leave 
28?     20?     12? 

16.  How  many  fours  must  be  added  to  20  to  equal  6 
fours  ?     8  fours  ? 

17.  How  many  fours  must  be  taken  from  16  to  leave 
3  fours  ?     1  four  ? 

18.  3  fours  +  7  =  ?  9  fours  —  1  =  ? 

19.  7  fours  +  3  =  ?  11  fours  -  2  =  ? 
Call  for  similar  questions  from  pupils. 

20.  How  many  fours  in  8  ?     In  2  eights  ?     3  eights  ? 

21.  (8  X  3)  -  4  =  ?     (8  X  5)  ^  4  =  ?     (8  x  6)  -^  4  =  ? 

See  note  after  Ex.  88,  p.  111. 

22.  How  many  quarters  of  a  dollar  equal  a  whole  dollar  ? 
How  many  quarters  equal  3  dollars  ?  5  dollars  ?  7  dollars  ? 

23.  How  many  quarters  of  a  dollar  equal  one  half  dollar  ? 
1  and  1-  dollars  1     2^-  dollars  ? 


156  FOURS 

24.  If  I  cut  6  apples  into  fourths,  to  how  many  boys  can 
I  give  one  fourth  of  an  apple  ? 

25.  To  how  many  boys  could  I  give  ^  of  an  apple  if  I 
had  5  apples  ?     7  apples  ?     9  apples  ?     10  apples  ? 

26.  HoAV  many  dollars  will  it  take  to  give  20  boys  a 
quarter  of  a  dollar  apiece  ?    36  boys  ?    48  boys  ?    24  boys  ? 

27.  How  many  pounds  of  coffee,  at  a  quarter  of  a  dollar 
a  pound,  can  be  bought  for  1 2  ?     For  1 3  ?     For  $5  ? 

28.  How  many  fourths  in  the  whole  of  anything  ? 

29.  How   many  fourths  in  two   whole   things  ?     In  4 
whole  ones  ?     3  ?     7  ? 

30.  How  many  fourths  in  2  wliole  ones  and  ^  ?  In  3|  ? 

31.  How  many  fifths  in  2i  ?  3f  ?  5|  ?  4^  ?  8^  ?  9f  ? 

32.  Which  is  greater,  and  how  much,  7  bu.  or  29  pk.? 
9  bu.  or  34  pk.?     18  pk.  or  5  bu.?     23  pk.  or  6  bu.? 

33.  How  mau}^  pk.  in  a  bushel  and  a  half  ?     In  2  bushels 
and  a  half?     In  3|^  bu.?     4^  bu.?     5-|  bu.?     6J  bu.? 

34.  What  is  the  ratio  of  a  quarter  of  a  dollar  to  2  dollars  ? 
A  peck  to  3  bushels  ?     A  quart  to  4  gallons  ? 

35.  #  =  ?     Ans.  2J-. 

36      1—9        11—?        _1J  —  '?        1_5  —  ?        i_a  —  ?        _1^  —  ? 
OD.     2  ~~  '  2    ~  '  2    ~  '  2  *  2~'  2  ■ 

5  —  ?  9  —  ?         11—?         _1  3.  _  ?         J_a  —  ?         2JL  —  ? 

^—  •  ^—  •  1[—  •  ¥~-  4~-  4~- 

37.  Divide:  4)897     4)9893     4)827     4)7389     4)6253 
Let  the  quotients  be  expressed  in  mixed  numbers. 

38.  Divide  by  4  each  of  the  numbers  between  200  and 
300  Avhose  unit  figure  is  7. 

39.  Use  4  as  a  divisor  of  each  of  the  numbers  between 
300  and  400  whose  unit  figure  is  5. 


FOURS 


157 


Fig.  1 


Fig.  2 


40.  How  many  equilateral  tri- 
angles in  Fig.  1  ?  Copy  it  by  plac- 
ing triangles.  Can  you  separate 
your  figure  into  three  equal  large 
triangles  ?  Each  large  triangle 
is  what  fractional  part  of  the 
whole  figure  ?  Each  small  tri- 
angle is  what  part  of  the  large 
triangle  to  which  it  belongs  ? 

41.  How  much  is  :|  of  J?    How  many  twelfths  in -J?   In -I? 

42.  Copy  Fig.  2  by  plac- 
ing triangles.  How  many 
triangles  in  it  ? 

43.  Each  triangle  is  what 
fractional  part  of  the  whole 
figure  ?  Show  J  of  the  figure.  Show  J  of  it,  and  tell 
how  many  12ths  |  equal.  Show  ^  of  it,  and  tell  how 
many  twelfths  ^  equals. 

44.  Separate  your  figure  into  6  rhombuses.  Each  rhom- 
bus is  what  fractional  part  of  the  whole  figure  ?  Each 
triangle  is  what  fractional  part  of  a  rhombus  ? 

45.  ^  =  how  many  12ths  ?     |  =  how  many  12ths  ? 

I  =  how  many  12ths  ?     ^  =  ho^\'  many  12thg  ? 

I  =  how  many  12ths  ?     What  is  j  of  }  ? 

If  the  children  cannot  see  the  facts  whicli  these  questions  are 
intended  to  bring  out,  do  not  let  them  memorize  them.  Come  back 
to  the  work  again  with  a  different  figure  and  lead  them  on  more 
slowly.  • 

46.  Copy  Fig.  2  again.  Separate  it  into  4  equal  trape- 
zoids. Each  trapezoid  is  what  fractional  part  of  the 
whole  figure  ?  Each  triangle  is  what  fractional  part  of 
a  trapezoid  ? 


158 


FOURS 


47.  ^  =  how  many  12ths  ?     -|  =  how  many  12ths  ? 
|.  =  how  many  12ths  ?     i  of  ^  =  what  ? 

48.  Put  the   figure   together   again.      Separate  it   into 
3  equal  parts.     How  many  triangles  in  each  part  ? 

49.  1  =  how   many   twelfths  ?      f  =  how  many  12ths  ? 
^  of  ^  =  what  ? 

50.  Draw^  a  circle  and  draw  a  line  across 
it,  dividing  it  into  halves. 

Show  pupils  how  to  draw  a  circle  by  the  aid  of 
dividers  or  a  string  or  a  slip  of  pasteboard  turning 
on  a  pin. 


Fig.  3 


51.    Divide   each   half   of   the    circle    into 
halves.     What  is  J  of  ^  ? 


Fig.  4 


52.    Divide  each  fourth  of  the  circle  into 
halves.     What  part  of  the  whole  is  J  of  ^  ? 


Fig.  5 


53.  Divide  each  eighth  of  the  circle  into 
halves.  How  manv  divisions  in  the  whole 
circle  ?     What  part  of  the  whole   is  J  of  ^  ? 

Let  these  circles,  large  and  bold,  be  drawn  upon  the 
board  and  left  there  for  some  time.  The  children 
should  discover  and  report  from  them  the  facts  called 
for  in  the  following  questions,  and  in  many  similar  ones.  Gradually 
discard  this  work  with  the  concrete,  and  lead  the  pupils  to  the  use  of 
figures  as  symbols. 

5*-     ^-*=?       J-f=?       i-*=? 

i+i=?     i+i=?     i+i-? 


Fig.  6 


FOURS  159 

55.    The  whole  circle  —  J-g  =  how  many  16ths  ? 


56.        i  -  Ve 


3   _   _ 

8  16 

!  +  -!-  = 
8   ^   16 


16 

„ —  '>  T 

■g"  16 

8     '     16         •  ¥    "^   16         • 

57.  Name  the  denominators  of  some  of  the  fractions  in 
Ex.  56. 

58.  In  a  written  fraction,  the  number  above  the  line  is 
called  the  Numerator.  Name  the  numerators  of  some  of 
the  fractions  in  Ex.  b6. 

59.  Write  a  fraction  with  an  odd  number  for  the 
numerator  and  an  even  number  for  the  denominator. 

60.  Write  a  fraction  with  4  as  the  numerator  and  8  as 
the  denominator,  and  tell  what  the  fraction  equals. 

61.  Write  a  fraction  with  4  as  the  numerator  and  12 
as  the  denominator,  and  tell  what  it  equals. 

62.  Write  a  fraction  with  5  as  the  numerator  and  10 
as  the  denominator,  and  show  what  it  equals. 

63.  Write  some  fractions  in  which  the  luimerator  is 
just  J  as  large  as  the  denominator,  and  show  what  each 
fraction  equals. 

64.  Write  some  fractions  in  which  the  denominator  is 
just  3  times  as  large  as  the  numerator,  and  show  what 
each  fraction  equals. 

65.  Write  some  fractions  with  4  as  the  denominator 
and  a  multiple  of  4  as  the  numerator,  and  tell  what  each 
fraction  equals. 

66.  What  is  i  of  12  ?     I  of  12  ? 

67.  What  is*  the  ratio  of  4  to  8  ?     4  to  12  ?     8  to  12  ? 
Use  chart  drill  as  suggested  in  note  after  Ex.  53,  pp.  117,  118. 

68.  There  are  12  apples  in  a  basket  and  |-  as  many  on 
a  plate.     How  many  apples  are  on  the  plate  ? 


160  FOURS 

69.  What  is  ^  of  16  ?     I  of  16  ?     |  or  J  of  16  ? 

70.  What  is  the  ratio  of  12  to  16  ?     Of  8  to  16  ? 

71.  If  John  has  16  marbles  and  James  ^  as  many,  how 
many  marbles  has  James  ? 

72.  The  price  of  4  marbles  is  what  part  of  the  price  of 
8  marbles  ?  How  much  will  4  marbles  cost  if  8  marbles 
cost  10  cents  ?    14  cents  ?    20  cents  ?    24  cents  ?   40  cents  ? 

73.  The  cost  of  4  marbles  is  what  part  of  the  cost  of 
12  marbles  ?  How  much  will  4  marbles  cost  if  12  mar- 
bles cost  15  cents  ?     24  cents  ?     30  cents  ?     18  cents  ? 

74.  What  part  of  a  pound  is  4  ounces  ?      8  ounces  ? 

75.  If  a  pound  of  candy  costs  40  cents,  what  part  of  the 
money  will  4  ounces  cost  ?    How  many  cents  will  they  cost  ? 

76.  How  much  will  8  marbles  cost  if  16  marbles  cost 
10  cents  ?     20  cents  ?     30  cents  ?     40  cents  ?     50  cents  ? 

77.  What  is  I  of  20  ?     |  of  20  ?     |  of  20  ?     |  of  20  ? 

78.  What  is  the  ratio  of  8  to  20?    16  to  20  ?    12  to  20  ? 

79.  If  a  yard  of  ribbon  costs  20  cents,  how  much  will 
^  of  a  yard  cost  ?     f  ?     ^?     f  ? 

80.  If  20  men  do  a  piece  of  work  in  a  day,  how  much 
of  the  work  is  done  by  4  men  ?     12  men  ?     8  men  ? 

81.  If  f  40  is  paid  for  the  work,  how  much  should  be 
paid  to  4  men  ?     12  men  ?     8  men  ?     16  men  ? 

82.  Fill  out  and  learn  the  following  : 

i  of  24  =  The  ratio  of    4  to  24  is  — 

I  or  J  of  24  =  The  ratio  of    8  to  24  is  — 

/     I  or  J-  of  24  =  The  ratio  of  12  to  24  is  — 

I  or  I  of  24  =  The  ratio  of  16  to  24  is  — 

I  of  24  =  The  ratio  of  20  to  24  is  — 

When  these  ratios  can  be  given  instantly,  give  and  call  for  many 
illustrative  problems. 


FOURS  161 

83.  How  many  hours  in  i  of  a  day  ?     In  |^  of  a  day  ? 

84.  What  part  of  a  day  is  12  hours  ?     8  hours  ? 

85.  John  bought  something  for  24  cents  and  sold  it  so 

as  to  gain  6  cents.     What  was  the  ratio  of  the  gain  to  the 
cost  ? 

86.  Take  the  number  28  and  make  a  table  showing  ^, 
f,  •••  -{  of  it. 

87.  Make  a  table  showing  what  ratio  each  of  the  mul- 
tiples of  4  less  than  28  has  to  the  number  28. 

88.  If  a  pound  of  candy  costs  28  cents,  how  much  will 
f  of  it  cost?     f?     f?     I?     5? 

89.  If  28  men  do  a  piece  of  work  in  a  week,  what  part 
of  it  is  done  by  4  men  ?     12  men  ?     8  men  ?     16  men  ? 

90.  If  they  are  paid  |>14  a  day,  how  much  will  4  men 
receive  for  each  day's  work  ?    8  men  ?    18  men  ?    24  men  ? 

91.  Take  32  and  make  a  table  showing  i,  |,  •••  |  of  it. 

92.  Make  a  table  showing  the  ratio  to  32  of  each  of  the 
multiples  of  4  that  are  less  than  33. 

Require  these  ratios  in  their  lowest  terms. 

93.  If  a  pound  of  candy  costs  32  cents,  how  much  will  | 
of  it  cost?     f?     I?     1?     1?     I? 

94.  If  32  cents  are  paid  for  some  candy,  how  much  of 
it  can  be  bought  for  4  cents  ?     12  cents  ?     28  cents  ? 

95.  A  flower  bed  is  8  ft.  long  and  4  ft.  wide.  Make  a 
picture  of  it.     How  many  sq.  ft.  in  ^  of  it  ?     In  |  of  it  ? 

96.  Make  a  table  showing  -J,  |,  ...  -|  of  36. 

97.  Make  a  table  showing  the  ratio  to  36  of  each  of  the 
multiples  of  4  that  are  less  than  38. 

98.  j\[r.  Smith  is  36  years  old.  How  old  is  his  son 
whose  age  is  |  of  Mr.  Smith's  age  ?  How  old  is  his 
daughter,  whose  age  is  |  or  -J  of  her  father's  ?     His  wife's 

HORN.  ARITH. 11 


162 


FOURS 


age  is  I  of  his,  how  old  is  she  ?  His  brother's  age  is  |  of 
his,  how  old  is  his  brother  ?  His  sister's  age  is  -J  of  his, 
how  old  is  his  sister  ? 

99.    A  bolt  of  cloth  contains  36  yd.      What  part  of  it 
is  20  yd.?     24  yd.?     32  yd.?     16  yd.?     12  yd.?     28  yd.? 

100.    If  the  whole  bolt  is  worth  $  18,  how  mucli  will  4 
yd.  cost  ?     8  ydo?     12  yd.?     20  yd.?     28  yd.?     32  yd.? 

A  B       101.    Fig,    7   rep- 

resents a  rectangle 
9  feet  long  and  4 
feet  wide.  How 
many  square  yards 
<-^  are  represented  by 
ABOD?  Draw  Fig. 
•^^^-  '  7    and    show    how 

many  square  yards  the  whole  figure  represents. 

102.  What  is  the  ratio  of  the  square,  ABOB^  to  the 
whole  figure  ? 

103.  Make  a  table  showing  ^  •••  i^  of  40. 

104.  Make  a  table  showing  the  ratio  to  40  of  each  of 
the  multiples  of  4  that  are  less  than  43. 

105.  A  certain  town   is  40   miles  from   New  Orleans. 


D 

How  far  from  the  town  is  a  man  who  has  traveled  -^  of 
the  distance  from  it  to  New  Orleans  ?  How  far  is  he 
from  New  Orleans  ? 

106.  When  he  has  traveled  -^^  of  the  way  to  New 
Orleans,  how  far  is  he  from  the  town  he  started  from  ? 
How  far  is  he  from  New  Orleans  ?  How  far  is  he  from 
each  place  when  he  has  traveled  -^^  or  J  of  the  way  ? 
^  of  the  way  ?     -^  of  the  way  ? 

107.  Find,  by  measuring,  how  many  gills  make  a  pint. 
How  many  gills  in  3  pints  ?     5  pt.  ?     9  pt.  ?     6  pt.  ? 


FOURS  163 

108.  Fill  out  and  learn  the  table  of  Liquid  Measure. 

gills  (gi.)  =  1  pint  (pt.) 

pints  =  1  quart  (qt.) 

quarts  =  1  gallon  (gal.) 

109.  How  many  gi.  in  a  qt.?     2  qt.?     5  qt.?     9  qt.? 

110.  How  many  gi.  in  IJ  pt.?  3^  pt.?  5|-  pt.? 
6^pt.?     71  pt.?     11  qt.?     -ifqt.? 

111.  How  much  is  4  multiplied  by  30?  20?  60? 
90?     70?     50?     80?     40?     120?     150? 

112.  How  much  is  40  multiplied  by  2  ?     7  ?     8  ?     12  ? 

113.  How  much   is  40   multiplied  by  30  ?     80  ?     60  ? 

114.  How  many  are  3  fours  and  J  of  4  ?  7  fours  and 
i  of  4  ?     9  fours  and  |  of  4  ? 

See  note  after  Ex.  91,  p.  128. 

115.  Thomas  has  4  marbles,  and  James  has  3|  times  as 
many.     How  many  marbles  has  James  ? 

116.  Make  story  problems. 

117.  5  times  ^  equals  liow  many  wliole  ones  and  fourths 
over  ? 

118.  How  many  whole  ones  and  how  many  fourths 
over  in  6  times  ^?  7  times  ^?  9  times  |^?  12  times 
f  ?  8  times  1?  11  times  f  ?  16  times  ^?  30  times  i? 
40  times  ^  ?     29  times  ^  ?     3  times  |^  ?     5  times  -|. 

How  many  square  inches  in : 

119.  A  rectangle  9  in.  long  and  4  in.  wide  ? 

120.  A  right  triangle  9  in.  long  and  4  in.  wide? 

121.  A  rectangle  4  in.  long  and  2J  in.  wide? 

122.  A  right  triangle  4  in.  long  and  3|  in.  wide  ? 

123.  A  rectangle  8  in.  long  and  i  in.  wide  ? 

124.  A  right  triangle  12  in.  long  and  ^  in.  wide? 


134  FOURS 

Give  each  child  inch -cubes,  and  lead  the  class  to  find  surfaces, 
edges,  and  angles. 

125.  Each  side  of  a  cube  is  called  a  face.  How  many 
faces  has  a  cube  ? 

126.  Show  some  parallel  lines  on  a  cube.  Show  per- 
pendicular lines. 

127.  If  you  were  to  paste  a  strip  of  paper  along  each 
edge  of  an  inch-cube  so  as  to  bind  the  edges,  how  many 
inches  long  would  all  the  strips  be  ? 

128.  How  many  right  angles  has  each  face  ?  How 
many  right  angles  have  all  the  faces? 

129.  Make  a  layer  of  inch -cubes  4  inches  long  and  2 
inches  wide.     How  many  cubes  in  it  ? 

130.  Cover  this  layer  with  another  layer  of  inch-cubes. 
How  many  cubes  in  the  whole  figure  ? 

131.  How  man}^  cubes  would  it  take  to  build  it  up 
3  layers  high  ?  To  build  it  5  layers  high  ?  7  layers  ? 
6  layers  ?     9  layers  ? 

132.  If  it  were  4  layers  high,  one  cube  Avould  be  what 
part  of  the  whole  figure  ? 

133.  Figures  like  these  you  have  built  are  called  Square 
Prisms.  Build  with  inch-cubes  a  prism  5  in.  long,  4  in. 
wide,  and  2  in.  high,  and  tell  how  many  cuV)es  in  it. 

134.  Build  a  prism  3  in.  long,  3  in.  wide,  and  2  in.  high. 
How  many  layers  of  cubes  are  in  it?  How  many  cubes 
in  each  layer? 

Let  the  children  use  the  cubes  to  build  prisms  until  they  are  able 
to  get  the  required  facts  by  means  of  their  mental  imagery,  then 
encourage  them  to  "  think  how  it  looks."  Do  not  let  illustrative 
work  become  formal  nor  take  the  place  of  thuiking. 


FOURS  165 

135.    Find  the  number  of  cubic  inches  in  each  of  the  fol- 
lowing prisms,  the  measurements  being  given  in  inches  : 


Length 

Width 

Height 

Length 

Width 

Height 

5 

3 

1 

4 

3 

3 

5 

2 

2 

8 

4 

2 

5 

4 

2 

8 

3 

2 

3 

4 

2 

4 

2 

9 

136.  How  many  cubic  inches  in  a  cube  each  edge  of 
which  is  2  in.  long?     3  in.?     4  in.?     5  in.? 

137.  How  many  square  inches  in  all  the  faces  of  a  2- 
inch  cube  ?     3-inch  cube  ?     4-inch  cube  ?     5-inch  cube  ? 

138.  An  inch  cube  equals  what  part  of  a  2-inch  cube? 
What  is  the  ratio  of  an  inch  cube  to  a  3-incli  cube  ?  To 
a  4-inch  cube  ?     To  a  5-inch  cube  ? 

139.  How  many  inch-cubes  can  you  put  into  a  box  that 
measures  on  the  inside  5  in.  long,  4  in.  wide,  and  3  in.  deep? 

140.  Estimate  the  length,  width,  and  depth  of  a  box, 

and  tell  how  many  cubic  inches  it  can  hold. 

Let  the  children  inclose  portions  of  space  with  blocks  and  tell  how 
many  cubic  inches  in  them. 

ft/ 

141.  How  many  cubic  feet  of  air  can  there  be  in  a 
closet  that  is  4  ft.  long,  3  ft.  wide,  and  9  ft.  high  ? 

142.  Multiply  444  by  each  of  the  numbers  that  are 
greater  than  2  and  less  than  10. 

Multiply      44  The  process  of    mnltiplying  by  a  number 

by      11         of   more    than  one    place    should    be    shown 

"~TT        simply  as  a  process  which  brings  the  desired 

result.     Later,  when  the  children  have  become 

'^^  expert   and   are  ready  for  the  insight,  show 

484         them  that  in  multiplying,  for  instance,  444  by 

111,  they  are  finding  the  sum  of  one  hundred 

444's,  ten  444's,  and  one  444. 


166  FOURS 

143.  Find  products: 

64  54  24  54  84  94 

n  n  11  1?  12  12 

144.  Multiply  14  by  itself. 

145.  Square  15,  21,  22,  23,  35,  38,  88,  89,  48,  49,  58,  59. 

146.  Multiply  444  by  each  number  between  10  and  20. 
Find  the  cost  of: 

147.  18  yd.  of  cloth  at  il.55  a  yd. 

148.  29  hats  at  f  1.45  apiece. 

149.  35  pounds  of  tea  at  il.l5  per  pound. 

150.  345  tons  of  coal  at  $8.25  per  ton. 

151.  There  are  2000  pounds  in  a  ton.  How  many 
pounds  in  40  tons?     In  70  tons?     IJ  tons?     2^  tons? 

152.  Add:   41     4J     3 J     51     6 J     9| 

2     41    4^    7|    8f    8| 

fi      !i      5i      ?i       ^       if 

153.  ]\Iary  wants  to  buy  for  her  mother  a  Christmas 
present  that  costs  f  3.00.  She  has  -t  1.22.  How  much 
does  she  lack  ? 

154.  She  saves  1^.35  more.  How  much  does  she  still 
lack? 

155.  She  earns  15  cents  a  week  for  7  weeks.  How 
much  does  she  still  lack  ? 

156.  She  saves  5  cents  a  week  for  9  weeks.  Does  she 
lack  any  then  ?     If  so,  how  much  ? 

157.  Find  (luotients:      4)3801       4)7897       9)8205 

158.  Find  I  of  8476,  8264,  8148,  9365. 

159.  One  of  the  girls  may  name  a  number  of  5  places, 
and  the  class  may  use  it  as  a  dividend  with  4. 

160.  Write  in  Arabic  notation  MDCCXCIX  and 
MDCCCXLIV,  and  find  their  difference. 


CHAPTER   XIII 
SEVENS 

Factors,  Compound  Fractions 


1 

SIUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

56 

6G 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1.  Begin  at  7  and  connt  by  sevens  to  98.  Practice 
until  you  can  count  quickly.  Ho^y  many  multiples  of  7 
are  less  than  100  ? 

2.  Begin  with  98  and  count  backAvards  by  7  to  0. 

3.  Write  and  learn  the  table  ending  '•■  12  times  7  are  84." 

4.  Name  in  order  all  the  multiples  of  7  that  are  less 
than  100  and  are  odd  numbers. 

167 


168  SEVENS 

5.  Name  in  order  all  the  even  mnltiples  of  7  that  are 
less  than  100. 

6.  Name  a  few  multiples  of  7  greater  than  100. 

7.  How  many  sevens  in  21  ?  84  ?  42  ?  56  ?  63  ?  28  ? 

8.  Give  the  5th  multiple  of  7  ;  7th,  9th,  11th,  4th,  6th. 

9.  Multiply  777    by   each  of   the   numbers   that   are 
greater  than  45  and  less  than  50. 

10.  If  a  bicycle  costs  157,   how  much  will    a   dozen 
bicycles  cost  ? 

11.  At  §87  apiece,  what  is  the  cost  of  24  bicycles  ? 

12.  At  i477  apiece,  how  much  would  13  pianos  cost? 

13.  Add  2  sevens  to  42,  63,  21,  49,  70,  14,  28,  56,  35. 

14.  Take  2  sevens  from  77,  42,  21,  56,  84,  63,  28,  49,  35. 

15.  How  many  sevens  must  be  added  to  28  to  make  42  ? 
5(j?  70?  84?  63?  35?  49?  77? 

16.  How  many  sevens  must  be  subtracted  from  84  to 
leave  70  ?  5(j  ?  77  ?  63  ?  49  ?  42  ?  28  ?  14  ?  35  ? 

17.  How  many  sevens  must  be  added  to  21  to  make 
4  sevens  ?     7  sevens  ?     9  sevens  ?     6  sevens  ?     8  sevens  ? 

18.  How  many  sevens  must  be  taken  from  77  to  leave 
9  sevens  ?     6  sevens  ?     8  sevens  ?     10  sevens  ?     5  sevens  ? 

19.  5  sevens4-3  =  ?     8  sevens +  5  =  ?     6  sevens  +  6  =  ? 

20.  8  sevens  — 5  =  ?     11  sevens  — 6  =  ?     7  sevens  — 5  =  ? 

21.  Which  multiple  of  7  is  35?  49?  77  ?  21  ?  56?  84? 

22.  Add  2  to  the  3d  multiple  of  7.     To  the  6th.     10th. 

23.  Subtract  4  from  the  2d  multiple  of  7.     From  the 

7th,  9th,  5th. 

24.  50  is  how  many  more  than  the  7th  multiple  of  7  ? 
How  many  less  than  the  8th  multiple  of  7  ? 

Call  for  similar  questions  from  pupils. 


SEVENS  169 

25.        7—   •  7~-  7'  7-  7*  7 

1  5   _  '>         3  8—9         6_6    _  9         89.  _  ?         3^L  —  '?         5_3  _  ? 
7—   •  7~~*  7~-  7'~-  7~'  7 

26.  At  7  cents  a  yard,  how  many  yards  of  ribbon  could 
you  buy  for  15  cents  ?     23  cents  ?     29  cents  ?     36  cents  ? 

27.  Make  story  problems. 

28.  Divide  by  7  each  of  the  numbers  between  900  and 
1000  whose  unit  figure  is  4. 

29.  At  i  7  apiece,  how  many  music  boxes  could  be  bought 
for  1500?     1800?     1900? 

30.  Multiply  797  by  each  of  the  even  numbers  between 
31  and  39. 

31.  Multiply  897  by  each  of  the  odd  numbers  between 
40  and  50. 

32.  What  number  must  7  be  multiplied  by  to  make  a 
product  of  m  ?  63  ?  84  ?  49  ?  42  ? 

33.  Numbers  that  make  a  product  are  called  Factors  of 
that  product.  Name  two  factors  that  make  14,  35,  77,  21, 
49,  63,  84,  42,  28,  56,  70. 

34.  8  is  a  factor  of  24.  Name  the  other  factor  that 
helps  8  to  make  24.  4  is  also  a  factor  of  24.  Name  the 
other  factor  that  helps  4  to  make  24. 

35.  2  is  one  of  a  pair  of  factors  that  help  each  other 
to  make  24.     What  is  the  other  factor  ? 

36.  Give  all  the  pairs  of  factors  that  you  can  of  12,  18, 
20,  30. 

37.  44  =  4  X  ?      25  =  5  X  ?      28  =  4  x  ?      42  =  7  x  ? 

38.  Write  all  the  multiples  of  8  that  are  less  than  48, 
and  the  pairs  of  factors  into  which  they  can  be  divided. 

Let  pupils  begin  with  the  smallest  factors  and  work  regularly ;  as, 

8  =  2x4; 
16  =  2  X  8  or  4  X  4 ; 
24  =  2  X  12  or  3  X  8  or  4  X  6. 

This  is  an  excellent  review  of  the  multiplication  table. 


170  SEVENS 

39.  Write  all  the  multiples  of  9  that  are  less  than  50, 
and  give  factors  of  them. 

40.  How  many  times  can  a  7 -inch  line  be  measured  off 
upon  a  21 -inch  line  ?  Upon  a  line  that  is  2  ft.  and  4  in. 
long  ?  Upon  a  line  that  lacks  an  inch  of  being  equal  to  a 
yard  ?  Upon  a  line  that  lacks  4  in.  of  being  5  ft.  long  ? 
Upon  a  line  that  is  1  yd.  1  ft.  and  1  in.  long  ? 

41.  How  many  rows  of  squares,  7  in  a  row,  make  42 
squares  ?     28  squares  ?     63  squares  ?     49  squares  ? 

42.  Place  or  draw  9  inch-squares  so  as  to  make  a  perfect 
square.     How  long  is  the  figure  ?     How  wide  is  it  ? 

43.  Draw  a  square  containing  49  square  inches  and 
mark  them  off.     Give  length  and  width  of  the  figure. 

44.  Draw  16  inch-squares  placed  in  a  perfect  square. 
How  long  is  each  side  of  the  square  ? 

45.  Draw  25  inch-squares  arranged  in  a  perfect  square. 
How  long  is  each  side  of  the  square  ? 

46.  What  is  the  square  of  3?     4?     5?     7? 

47.  Find  the  square  of  17  ;  27  ;  77  ;  97  ;  47. 

48.  What  number,  multiplied  by  itself,  will  give  9  ? 
25  ?     16  ?     100  ? 

49.  Give  other  numbers  that  are  made  of  two  equal  fac- 
tors. 

50.  A  number  that  is  made  of  two  equal  factors  is  called 
a  Perfect  Square,  and  each  of  the  equal  factors  is  called  a 
Square  Root  of  the  number.  What  is  the  square  root 
of  49  ?     25  ?     81  ?     16  ? 

51.  Write  some  other  numbers  that  are  perfect  squares, 

and  give  their  square  roots. 

Oral  class  exercises  like  the  following  are  useful :  "  Take  the  square 
root  of  9,  double  it,  add  4,  divide  by  2,  square,  subtract  5,  take  |,  add," 
etc. 


SEVENS  171 

52.  The  floor  of  a  square  room  contains  49  sq.  ft.  How 
long  is  one  side  of  the  room  ?  How  many  feet  around  all 
the  edges  of  the  floor  ? 

53.  How  many  feet  around  the  edge  of  the  floor  of  a 
closet  if  the  floor  is  square  and  contains  25  sq.  ft.? 

54.  How  many  feet  around  the  edge  of  a  square  floor 
that  contains  16  sq.  yd.?     25  sq.  yd.?     6-4  sq.  yd.? 

55.  How  many  feet  around  a  room  11  ft.  long  and  7  ft. 
wide  ? 

56.  How  much  will  a  pt.  of  oil  cost  at  7  cents  a  gi.? 
At  9  cents ''     3  cents  ?     8  cents  ? 

57.  If  it  takes  you  7  minutes  to  Avalk  to  school,  how 
many  minutes  do  you  spend  in  walking  to  school  each 
day  ?  How  many  in  a  week  of  5  school  days  ?  In  a 
school  month  or  20  days  ? 

58.  At  7  cents  a  qt.,  how  much  will  a  gal.  of  milk  cost  ? 

59.  How  much  will  3  yd.  of  wire  cost  at  7  cents  a  ft.  ? 
At  7  cents  an  in.  ? 

60.  7  times  7  in.  =  how  much  more  than  4  ft.  ? 

61.  How  much  will  2  qt.  and  1  pt.  of  berries  cost  at  7 
cents  a  pt.  ?     At  14  cents  a  qt.  ? 

62.  How  many  apples  would  it  take  to  give  6  boys  7 
apples  apiece  ? 

63.  How  many  marbles  must  you  have  to  give  7  marbles 
to  each  of  8  boys? 

64.  How  many  days  in  4  weeks  ?  5  Aveeks  ?  8  weeks  ? 
6  weeks  ?     3  weeks  and  4  days  ?     5  weeks  and  1  day  ? 

65.  What  part  of  a  week  is  1  day  ?     3  days  ?     5  days  ? 

66.  HoAv  many  weeks  and  sevenths  of  a  week  in  11 
days?  22  days?  30  days?  36  days?  44  days?  50 
days  ?     69  days  ?     82  days  ? 


172  SEVENS 

67.  How  many  weeks  in  583  days  ?  687  days  ?  599 
days  ?     1601  days  ? 

68.  How  many  inch  cubes  will  it  take  to  make  a  square 
prism  7  in.  long,  3  in.  wide,  and  2  in.  high  ? 

69.  How  many  cubic  inches  in  a  box  7  in.  long,  4  in. 
wide,  and  2  in.  deej)  ? 

70.  How  many  cubic  feet  of  space  in  a  closet  7  ft.  long, 
6  ft.  wide,  and  8  ft.  high  ? 

71.  How  many  cubic  feet  of  space  in  a  closet  7  ft. 
square  and  0  ft.  high  ? 

72.  Arrange  7  equilateral  triangles  as  in  Fig.  1.     Can 

you  take  aAvay  two  trapezoids  from  Fig. 
1  and  leave  one  triangle  ? 

73.    What  is  the  ratio  of  one  triangle 

to  the  whole  figure  ?     Of  one  trapezoid 

Pj^  "I^  to  the  whole  figure  ?    Of  two  trapezoids 

to  the  whole  figure  ? 

74.    Copy  Fig.  2.     Take  away  two  trapezoids  and  show 

what  is  left. 

75.  Can  you  show  how  Fig.  1  may  be 
changed  into  Fig.  2  by  changing  the 
position  of  one  trapezoid  ? 

76.  How  long  would  the  perimeter  of 
Fig.  2  be  if  each  side  of  a  triangle  were 

^'^-  ^  7  in.  ?     8  in.  ?     9  in.  ?     5  in.  ?     11  in.  ? 

77.  Take  away  f  of  Fig.  2.  Tell  how  long  the  perim- 
eter of  the  figure  that  is  left  would  be  if  each  side  of  the 
triangles  were  7  in.  long. 

78.  How  many  sevenths  make  the  whole  of  anything  ? 

79.  How  many  sevenths  in  2  whole  ones?     9?     5?     7? 

80.  How  many  7ths  in  2|  ?    3f  ?    5f  ?     7|  ?     8 f  ?     9f  ? 


SEVENS 


173 


Fig.  3 


Fig.  4 


Fig.  5 


81.  Copy  Fig.  3  by  placing 
equilateral  triangles.  Each  tri- 
angle is  what  part  of  Fig.  3  ? 

82.  Divide  each  equilateral  tri- 
angle into  two  right  triangles  as 
in  Fig.  4.  Each  right  triangle  is 
what  part  of  the  whole  figure  ? 
J  of  ^  =  ? 

83.  Copy  Fig.  5  by  placing 
equilateral  triangles.  1  triangle 
is  what  part  of  Fig.  5  ? 

84.  Separate  your  copy  into  halves.     1  triangle  is  what 

part  of  a  half  of  the  figure  ?     i  of  ^  =  ? 

Show  pupils  that  "when  we  wish  to  find  the  value  of  a  compound 
fraction,  instead  of  dividing  and  subdividing  an  object  and  counting 
the  parts,  we  multiply  the  numerators  of  the  fractions  together  for  a 
new  numerator,  and  the  denominators  for  a  new  denominator.  Let 
them  try  the  plan  with  some  small  fractions  and  prove  its  correctness 
by  building  figures  and  separating  them  into  parts. 

85.     |0fl  =  ?  J0f3=?  i0fi  =  ?  i0ii  =  ? 

86.  A  fraction  of  a  fraction  is  called  a  Compound  Frac- 
tion. Write  some  compound  fractions  and  find  their 
values. 

87.  }0ii  =  ?         |0f  J  =  ?  |ofVo=?         }0ii  =  ? 

88.  |of  J  =  Jj.     \¥hat  will  f  of  1  equal  ? 
How  much  are  f  of  ^ ?     -fofj?     fofj?     iof|? 
ioii=:?     ioff=?     |off  =  ?     foff  =  ? 
I  of  |-  =  ?     (Cancel  when  you  can.)     |-  of  ^^  =  ? 

92.    What  is  the  ratio  of  7  to  14  ?     7  to  21  ?     14  to  21  ? 
28  to  21  ?     35  to  21  ? 
Use  chart  drill. 


89. 
90. 
91. 


174  SEVENS 

93.  If  a  yard  of  ribbon  costs  21  cents,  how  much  ribbon 
can  you  get  for  7  cents  ?    14  cents  ?    28  cents  ?    42  cents  ? 

94.  What  is  the  ratio  of  7  to  28  ?    21  to  28  ?    14  to  28  ? 

95.  If  28  ajDples  cost  10  cents,  how  much  will  14 
apples  cost  ? 

96.  A  bed  of  pinks  2  feet  square  is  a  part  of  a  floAver 
bed  7  ft.  long  and  4  ft.  wide.  Make  a  picture  of  them, 
drawing  to  a  scale  of  1  inch  to  a  foot.  What  is  the  ratio 
of  the  bed  of  pinks  to  the  whole  flower  bed  ? 

97.  How  much  is  i  of  35  ?     -|  ?     |  ? 

98.  35-iof35  =  ?     35-fof35  =  ?    35--|of35=? 

99.  What  part  of  35  is  28  ?     14?     21?     42? 

100.  Mary  had  35  cents  and  spent  ^  of  them.  How 
many  had  she  left  ? 

101.  Mr.  Baker  borrowed  f  35.  When  he  had  paid  | 
of  it,  how  many  dollars  did  he  still  owe  ? 

102.  John  had  35  marbles,  and  Albert  had  ^  as  many. 
How  many  had  Albert  ? 

103.  Mr.  Lane's  watch  chain  is  worth  -|  as  much  as  his 
watch,  which  is  worth  f  35.  How  much  is  the  chain 
worth?  How  much  are  they  both  worth?  The  watch 
is  worth  how  much  more  than  the  chain  ? 

104.  Fill  out  and  learn  the  following: 

1  of  42  =  The  ratio  of to  42  is 

f  or  ^  of  42  =  The  ratio  of to  42  is 

f  or  1  of  42  =  The  ratio  of to  42  is 

f  or  I  of  42  =  The  ratio  of to  42  is 

f  of  42  =  The  ratio  of to  42  is 

f  of  42  =  The  ratio  of to  42  is 


SEVENS  175 

105.  Fill  out  the  following  table  of  Avoirdupois  Weight : 

— —  ounces  (oz.)  =  1  pound  (lb.). 
pounds  =  1  Ton  (T.). 

106.  How  many  oz.  in  2  lb.  and  3  oz.  ?     2  lb.  7  oz.  ? 

107.  A  jar  contains  2  lb.  and  10  oz.  of  l)utter.  How 
many  oz.  in  the  jar  ?  How  many  oz.  in  ^  of  it  ?  |  of  it  ? 
lofit?    fofit?    lofit? 

108.  There  are  42  gal.  of  oil  in  a  barrel.  When  i  of 
the  oil  is  drawn  out,  how  many  gal.  are  left  ?  How  many 
are  left  when  |  or  ^  of  the  oil  is  drawn  out  ?  When  |-  or 
I  is  drawn  out  ? 

109.  Grace  wishes  to  buy  a  doll  that  costs  42  cents. 
She  has  35  cents.  What  part  of  the  price  has  she,  and 
how  many  sixths  of  the  price  does  she  need  ? 

110.  Make  a  table  showing  the  7ths  of  49  from  ^  to  ^. 

111.  Make  a  table  showing  the  ratio  of  each  multiple 
of  7  that  is  smaller  than  50,  to  49. 

112.  Mr.  Jones  can  do  a  piece  of  work  in  49  days. 
How  many  days  will  it  take  him  to  do  ^  of  it  ?  How 
many  Aveeks  ?  How  many  days  will  he  need  to  do  |^  of  it  ? 
1^  ?    1^  ?    ^  ?     How  many  weeks  in  each  case  ? 

113.  There  are  49  yards  in  a  bolt  of  cloth.  What  frac- 
tional part  of  it  remains  after  7  yd.  of  it  are  sold  ?  After 
21  yd.  are  sold  ?     After  35  yd.  are  sold  ? 

114.  Gertrude's  age  is  f  of  49  years.  Her  father's  age 
is  ^  of  49  years,  and  her  mother's  age  is  |-  of  49  years. 
How  old  is  each  one  of  the  family  ? 

115.  How  much  younger  is  Gertrude  than  her  father  ? 
Than  her  mother  ? 

116.  Gertrude's  age  is  in  what  ratio  to  her  father's  age? 
To  her  mother's  age  ? 


176  SEVENS 

117.  Make  a  table  showing  the  8ths  of  56  from  ^  to  |. 

118.  Make  a  table  showing  what  ratio  each  multiple  of 
7  that  is  smaller  than  G-l  has  to  56. 

119.  Mary  spent  56  days  in  a  visit  to  her  aunt  at  St. 
Louis.  When  she  had  been  there  a  week,  what  part  of 
her  visit  was  past  ?  What  part  of  it  was  past  when  she 
had  been  there  21  days  ?     35  days  ?     49  days  ? 

120.  A  farmer  brought  56  lb.  of  butter  to  market  and 
sold  I  of  it.     How  many  lb.  had  he  left  ? 

121.  He  received  $  14  for  his  56  lb. ;  how  much  did  he 
receive  for  28  lb.  of  it  ? 

122.  Anna  has  56  cents.  How  much  will  she  have 
when  she  has  spent  ^  of  it  ?     |  of  it  ?     |  of  it  ?     |  of  it  ? 

123.  Make  a  table  showing  the  9ths  of  63  from  J  to  ■^-. 

124.  Make  a  table  showing  what  fractional  part  of  63 
each  multiple  of  7  is  that  is  less  than  75. 

125.  A  gentleman's  house  is  63  miles  from  Denver. 
When  he  has  traveled  -J  of  the  way  to  Denver,  how  far  is 
he  from  his  own  house  ?  How^  far  from  Denver  ?  Tell 
how  far  he  is  from  each  place  when  he  has  traveled  -|  of 
the  way.     |.     f .     -J.     |. 

126.  A  garden  is  63  ft.  long  and  42  ft.  wide.  What  is 
the  ratio  of  its  width  to  its  length  ? 

127.  Make  a  tal)le  showing  lOths  of  70  from  -^  to  \^. 

128.  Make  a  table  showing  what  fractional  part  of  70 

each  multiple  of  7  is  that  is  less  than  75. 

129.  Arthur,  William,  and  Thomas  gave  70  cents  in 
charity.  Arthur  gave  -^^^  or  ^,  of  the  money,  William 
gave  y^^,  and  Thomas  gave  -f^,  or  |.  How  many  cents 
did  each  give  ? 


SEVENS  177 

130.  John  had  14  cents,  and  wished  to  buy  a  70-cent  cap. 
What  part  of  the  cost  of  the  cap  had  he  ?  What  part  did 
he  lack  ?  Wliat  part  did  he  lack  when  he  had  gained 
21  cents  more  ? 

131.  If  one  fan  costs  7  cents,  how  many  fans  can  be 
bought  for  56  cents  ?     28  cents  ?     63  cents  ?     84  cents  ? 

132.  If  2  fans  cost  14  cents,  how  much  will  3  fans  cost  ? 
5  fans  ?     7  fans  ?     4  fans  ?     8  fans  ? 

133.  How  much  will  8  fans  cost  if  7  fans  cost  21  cents  ? 
63  cents  ?     49  cents  ?     84  cents  ?     28  cents  ?     56  cents  ? 

134.  How  much  will  9  fans  cost  if  7  fans  cost  21  cents  ? 
49  cents  ?     63  cents  ?     14  cents  ?     28  cents  ?     56  cents  ? 

135.  What  is  the  product  of  70  multiplied  by  8  ?  6  ?  11  ? 

136.  What  is  the  product  of  7  multiplied  by  30  ?  60  ?  90  ? 

137.  How  much  are  2  sevens  and  |  of  7  ?  3  sevens 
+  f  of  7  ? 

Let  pupils  practice  giving  quickly  the  products  found  by  multiply- 
ing 7  by  each  of  the  smaller  integers  +  i,  f,  f,  etc. 

138.  How  many  sq.  in.  in  a  rectangle  7  in.  long  and 
5|  in.  wide  ?  How  many  in  a  right  triangle  of  the  same 
length  and  width  ? 

139.  Multiply       9      27      77  7  7      247      328 

by     10      10      10      100      1000        20        40 

140.  Tell  how  you  multiply  a  number  by  10  or  100. 

141.  Show  how  you  multiply  a  number  by  20,  30,  or 
any  multiple  of  10. 

142.  In  a  right  triangle  one  of  the  sides  that 
makes  the  right  angle  is  called  the  Base  and  the  -I 
other  the  Altitude.     Find  the  area  of  a  right  tri-  § 
angle  whose  base  is  7  inches   and   altitude   10  ^ 

inches.  Base 

HORK.    ARITH.  —  12 


178  SEVENS 

143.  By  holding  a  paper  triangle  in  different  positions, 
show  that  the  same  line  may  sometimes  be  called  the  base 
and  sometimes  the  altitude. 

144.  Draw  right  triangles,  making  the  bases  and  alti- 
tudes of  any  measurements  you  choose,  and  find  the  areas. 

145.  Add: 


7I 

9f 

6f 

H 

^ 

n 

^ 

4f 

4f 

H 

If 

3f 

If 

n 

6f 

^ 

9f 

H 

2^ 

8f 

n 

146.  There  are  4|  yards  of  cloth  in  May's  dress,  and 
2|  yards  in  her  jacket.  How  many  yards  are  there  in 
both  ? 

147.  From  7|-         35f       16f       27f       25^       64f      34 
take     4         Hi       _^       JH;         ly         2f      1} 

148.  For  five  years  Mr.  Smith's  family  saved  money  to 
raise  the  sum  of  $1000,  to  send  John  to  college.  The 
first  year  they  saved  §85.75,  the  second  year  f  98.50,  the 
third  year  ^f  197.50.  How  much  more  was  needed  to 
make  up  the  1 1000  ? 

149.  In  the  fourth  year  they  laid  aside  $195.45,  and 
John's  uncle  sent  him  a  check  for  $  150.  How  much 
more  was  needed  ? 

150.  In  the  fifth  year  they  saved  $245.50,  and  John 
earned  the  rest  in  vacation.     How  much  did  John  earn  ? 

151.  If  a  boy  earns  $3.95  a  week,  how  much  will  he 
earn  in  17  weeks  ?     In  19  weeks  ? 

What  will  be  the  cost  of  : 

152.  19  tons  of  hay  at  $9.75  per  ton  ? 

153.  23  barrels  of  flour  at  $4.75  per  barrel  ? 

154.  24  copies  of  "Alice  in  Wonderland"  at  $1.27  per 
copy  ? 

155.  28  bu.  of  wheat  at  $  .87  a  bu.? 


SEVENS  179 

Find  cost  of  each  article  if  : 

156.  3  tons  of  hay  cost 'f  24.75. 

157.  5  barrels  of  flour  cost  -i^  26.25. 

158.  7  geographies  cost  f  4.55. 

159.  9  bushels  of  wheat  cost  88.19. 

Let  the  children  find  the  actual  prices  of  various  articles  in 
common  use  in  their  locaUty,  and  bring  in  problems  based  upon 
them. 

160.  Name  two  factors  of  the  3d  multiple  of  7.  Of 
the  5th.     9th.     lltli.     8th.      6tli.     3d.     7th.     12th. 

161.  Of  wliat  number  is  7  the  square  root  ?  How 
long  is  one  side  of  a  square  that  contains  49  square 
inches  ? 

162.  What  number  that  has  7  for  a  factor  is  nearest  to 
20  ?     30  ?     50  ? 

163.  How  much  greater  is  tlie  product  of  7  and  8  than 
their  sum  ? 

164.  Thomas  went  fishing  at  7  o'clock  in  the  morning, 
and  came  home  7  hours  later.  At  what  time  did  he  come 
home  ? 

165.  Write  in  Arabic  notation  CCCCXXVII  and  VH, 
and  find  their  quotient. 

166.  Write  in  Roman  notation  the  present  year,  the 
year  in  which  you  were  born,  the  year  in  which  the 
Declaration  of  Independence  was  signed. 


CHAPTER   XIV 

SIXES 

Rod,   Hexagon,   Ixterest 


NUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

56 

66 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1.  Add  6  to  0,  and  keep  on  adding  sixes  until  you  have 
12  sixes.  What  number  have  you  ?  Which  multiple  of  6 
is  it  ? 

2.  Begin  at  the  12th  multiple  of  6  and  count  backwards 
by  sixes  until  nothing  is  left. 

3.  Learn  the  table  of  sixes  as  far  as  "  12  times  6  are  72." 

180 


SIXES  181 

Give  rapid  drill  on  6's  by  aid  of  this  figure. 

4.  How  nifiny  sixes  in  24  ?     42  ? 
54  ?     66?     36  ?  "  48  ?     72  ?     30  ? 

5.  Add  2  sixes  to  18.     30.     48.    \  ^ 
60.     54.     24. 

6.  Subtract  2  sixes  from  48.     60. 
24.     36.     66. 


11 


7.  How  many  sixes  must  be  added  to  12  to  equal  30  ? 
36?     24?     18?     42? 

8.  How  many  sixes  must  be  taken  from  48  to  leave  5 
sixes  ?     7  sixes  ?     4  sixes  ?     6  sixes  ? 

9.  Read  and  give  quotients  quickly: 

183060364872246642       54 
6666666666 

10.  If  you  put  36  square  inches  into  a  perfect  square, 
how  long  would  one  side  of  the  figure  be  ?  How  long 
would  the  perimeter  of  the  figure  be  ? 

11.  Of  what  number  is  6  the  square  root  ? 

12.  Multiply  6666  by  the  square  root  of  36  ;  by  the 
square  root  of  49  ;  by  the  square  root  of  64;  by  the  square 
root  of  81. 

13.  Use  6666  as  a  multiplicand  with  each  of  the  numbers 
greater  than  55  and  less  than  63. 

14.  Use  6  as  a  divisor  with  each  of  the  numbers  between 
500  and  600  whose  unit  figure  is  9. 

15.  What  factor  helps  6  to  make  18  ?  30?  54?  72?  60? 

16.  Name  two  numbers,  neither  of  which  is  6,  that  mul- 
tiplied together  give  18.     30.     24.     72.     12.     36. 

17.  Write  as  many  sets  of  factors  of  24  as  you  can.  Of 
26.     48.     30.     28.     40.     32.     20.     16. 


182  SIXES 

18.  Write  the  multiples  of  6  as  far  as  72,  and  give  the 
sets  of  factors  into  which  they  can  be  resolved,  as: 

6  =  2x3; 

,    12  =  2x6  or  3x4; 

18  =  2  X  9  or  3  X  6  ; 

24  =  2  X  12  or  3  X  8  or  4  X  6. 

Let  the  children  select  composite  numbers,  and  call  on  the  class  for 
the  factors  of  them.  Factoring  is  very  useful  in  helping  children  to 
see  the  relations  of  numbers,  and  is  not  difficult  for  them  if  they  know 
the  multiplication  tables. 

19.  Name  some  numbers  that  are  made  of  two  equal 
factors,  and  give  the  factors. 

20.  Fill  out  the  blanks  in  the  following  sentence : 
"  One  of  the  two  equal  factors  that  make  a  number  is 
called  the of  that  number." 

21.  15  X  15  =  225.     What  is  the  square  root  of  225  ? 

22.  Of  Avhat  number  is  16  the  square  root  ?  21  ?  23  ? 
18? 

23.  How  many  lb.  of  sugar  could  be  bouglit  for  48 
cents  when  sugar  is  6  cents  a  lb.  ?  8  cents  a  lb.  ?  4  cents 
alb.? 

24.  To  how  many  boys  could  you  give  6  marbles  apiece 
if  you  had  36  marbles  ?     66?     42?     30? 

25.  If  there  were  6  peas  in  a  pod,  how  many  peas  in 
9  pods?     7?     12?     8?     6? 

26.  Mrs.  Adams  cut  the  pies  at  a  picnic.  She  cut  7  pies 
into  sixths,  and  they  were  all  eaten  up.  Each  person  had 
one  piece  of  pie.     How  many  persons  were  at  that  picnic? 

27.  Anna  has  3  times  as  much  money  as  Mary,  who  has 
6  cents.     How  much  have  both  ? 

28.  John  has  5  times  as  much  money  as  James,  who  has 
6  cents.     How  much  more  has  John  than  James  ? 


SIXES  183 

29.  Five  boys  started  a  game  of  marbles.  Fred,  one  of 
the  boys,  had  no  marbles,  and  so  each  of  the  other  boys 
lent  him  6  marbles.    How  many  had  Fred  to  start  with  ? 

30.  Fred  won  35  marbles.  After  he  had  paid  his  play- 
mates, how  many  did  he  have  left  ? 

31.  What  is  the  ratio  of  6  to  12  ?     6  to  18  ?     12  to  18  ? 

32.  The  cost  of  12  apples  is  what  part  of  the  cost  of  18 
apples  ?  What  would  12  apples  cost  if  18  apples  cost  30 
cents  ?     15  cents  ?     21  cents  ?    24  cents  ?     9  cents  ? 

33.  What  is  the  ratio  of  6  to  24  ?  Of  12  to  24  ?  Of 
18  to  24  ? 

34.  How  much  is  1  of  24  ?     f  of  24  ?     |  of  24  ? 

35.  How  much  will  12  fans  cost  if  24  fans  cost  f  .50? 
1.60?     i.80? 

36.  How  many  hours  is  it  from  Cj  A.M.  on  Monday  till 
6  A.M.  on  Tuesday  ? 

37.  From  6  a.m.  till  noon  is  wliat  part  of  24  hours? 

38.  From  6  a.m.  till  6  p.m.  is  what  part  of  24  hours? 

39.  From  6  A.M.  till  midniglit  is  what  part  of  24 
hours? 

40.  What  is  the  ratio  of  G  to  30  ?  How  much  is  f  of 
30?     I  of  30?     I  of  30? 

41.  What  part  of  the  price  of  30  apples  is  the  price  of 
6  apples?     24  apples?     18  apples?     12  apples? 

42.  If  20  cents  are  paid  for  30  apples,  how  much  Avill  6 
apples  cost?     12  apples?     24  apples?     18  apples? 

43.  Fill  out  and  learn  the  following : 

i  of  3G  =  The  ratio  of  —  to  36  is  — 

I  or  J  of  36  =  The  ratio  of — -to  36  is  — 

J  or  |-  of  36  =  The  ratio  of  —  to  36  is  — 

J  or  I  of  36  =  The  ratio  of —  to  36  is  — 

1^  of  36  =  The  ratio  of  —  to  36  is  — 


184  SIXES 

44.  What  part  of  the  price  of  36  hats  is  the  price  of  6 
hats  ?     18  hats  ?     30  hats  ?     24  hats  ?     12  hats  ? 

45.  How  many  inches  in  J  of  a  yard  ?     In  |-  ?     -i  ?     |  ? 

46.  To  a  line  a  yard  long  what  is  the  ratio  of  a  line  6 
in.  long  ?  Of  a  line  1  ft.  long  ?  Of  a  line  a  foot  and  a 
half  long  ?     Of  a  line  2  ft.  long  ?     Of  a  line  1^  ft.  long  ? 

47.  Make  a  table  showing  7ths  of  42  from  ^  to  -|. 

48.  Express  the  ratio  to  42  of  each  of  the  multiples  of 
6  less  than  50. 

49.  42  gallons  of  oil  are  in  a  barrel.  Tell  what  frac- 
tion of  it  is  gone  and  what  fraction  of  it  is  left  when  12 
gal.  have  been  drawn  out.     24  gal.     30  gal.     36  gal. 

50.  If  the  whole  barrel  was  w^orth  $  28,  how  much  would 
6  gal.  cost  ?     12  gal.  ?     24  gal.  ?     30  gal.  ?     36  gal.  ? 

51.  Make  a  table  of  the  8ths  of  48  up  to  |. 

52.  Express  the  ratio  that  each  of  the  multiples  of  6 
less  than  61  has  to  48. 

53.  How  many  hours  from  12  o'clock  noon  on  Monday 
to  12  o'clock  noon  on  Wednesda}'? 

54.  From  12  o'clock  till  6  p.m.  on  Monday  is  what 
part  of  48  hours  ? 

55.  From  noon  till  midnight  is  what  part  of  48  hours  ? 

56.  From  midnight  till  6  p.m.  is  what  part  of  48  hours  ? 

57.  From  6  a.m.  Monday  to  6  p.m.  Tuesday  is  what 
part  of  48  hours  ? 

58.  From  6  A.M.  Monday  till  noon  on  Tuesday  is  what 
part  of  48  hours  ?  Make  a  table  showing  9ths  of  54  up 
to  JJ- 

59.  Express  the  ratio  that  each  of  the  first  10  multiples 
of  6  has  to  54. 


SIXES  185 

60.  The  price  of  0  marbles  is  what  part  of  tlie  price  of 
54  marbles?  If  54  marbles  cost  18  cents,  how  much  will 
6  marbles  cost  ?     18?     30?     12?     24?     48?     42?     36? 

61.  60  seconds  make  a  minute.  How  many  seconds  in 
2  minutes  ? 

Give  the  children  ideas  of  minutes  by  requhing  them  to  keep  per- 
fectly still  for  one  minute  by  the  watch,  and  of  seconds  by  having 
them  beat  time,  one  beat  to  a  second. 

62.  Fill  out  and  learn  the  table  of  Time  : 

seconds  (sec.)s=  1  minute  (min.). 

minutes  =  1  hour  (hr.). 

hours  =  1  day  (da.). 

days  =  1  week  (wk.). 

63.  How  many  sec.  in  3  min.?  5  min.?  9  min.?  4 
min.?     8  min.?     11  min.?     7  min.?     6  min.?     12  min. 

64.  How  many  sec.  in  8  min.  and  2  sec?  4  min.  and  7 
sec?     9  min.  and  3  sec?     8  min.  and  10  sec? 

65.  Make  a  table  showing  lOths  of  60  up  to  ^^. 

66.  Make  a  table  showing  the  ratio  to  60  of  each  of  the 
first  12  multiples  of  6. 

67.  What  part  of  a  nun.  is  12  sec?  30  sec?  18  sec? 
54  sec?     42  sec?     36  sec?     48  sec? 

68.  How  many  min.  in  2  hr.  and  8  min.?  5  hr.  and 
9  min.?     6  hr.  and  25  min.?     7  hr.  and  48  sec? 

69.  How  many  minutes  in  ^  an  hour  ?    ^?    ^?    J-?    |? 

i  ?      2  ?      _i_  ?      __3_  ?      JL.  9      _9_  ? 
3  •        3  •        10  •        10  •        10  •        10  • 

70.  How  many  minutes  in  3 J  hr.?  5^  hr.?  7^  hr.? 
8fhr.?     43^^hr.?     Q-^hv.?     10-^%  hr.? 

71.  What  part  of  an  hour  is  15  minutes?  30?  45?  10? 
5?     6?     24?     54?     36?     48?     42?     18? 


186  SIXES 

72.  Draw  a  line  on  the  floor  5i  yards  long.  How  many 
feet  long  is  it  ?  It  is  1  rod  long.  How  many  feet  in  2 
rods  ?     3  rods  ?     4  rods  ?     7  rods  ?     11  rods  ? 

Let  rods  be  measured  off  in  the  yard  or  upon  the  pavement  by- 
means  of  a  string  5|  yards  long,  and  give  practical  questions  in  meas- 
urement. 

73.  How  many  feet  in  the  perimeter  of  an  equilateral 
triangle,  each  side  of  which  is  1  rod  long  ? 

74.  Make  a  table  showing  lltlis  of  66  up  to  ^^. 

75.  Express  the  ratio  of  each  of  the  first  12  multiples 
of  6  to  66. 

76.  Robert's  house  is  66  rods  from  the  schoolhouse. 
When  he  has  gone  6  rods  on  his  way  to  school,  what  part 
of  the  distance  has  he  gone,  and  what  part  has  he  yet  to  go  ? 

77.  What  part  of  the  distance  has  he  gone,  and  what 
part  has  he  to  go  when  he  has  gone  18  rods  ?  30  ?  42  ? 
48  ?     80  ? 

78.  Draw  on  the  floor  a  square  that  is  1  rod  long  each 
way.     How  many  feet  in  its  perimeter  ? 

79.  How  many  square  rods  in  a  garden  that  is  9  rods 

long  and  (3  rods  wide  ?     8  rods  long  and  7  rods  wide  ? 

Let  the  children  mark  off  on  the  playground  a  jDieee  of  ground  a 
number  of  rods  long  and  a  number  of  rods  wide,  and  find  the  number 
of  square  rods. 

80.  320  rods  make  a  mile.  How  many  rods  in  4  miles  ? 
9  miles  ?     12  J  miles  ? 

Call  on  pupils  to  mention  places  that  are  about  a  mile  distant. 

81.  Fill  out  and  learn  the  table  of  Long  Measure  : 

inches  (in.)  =  1  foot  (ft.). 

feet  =  1  yard  (yd.). 

\  =  1  rod  (rd.). 
yards  j  ^ 

rods  =  1  mile  (mi.). 


SIXES  187 

82.  John  lived  a  mile  south  of  the  post  office.  His 
cousin  Henry  lived  2i  mi.  south  of  it.  How  many  miles 
must  John  walk  to  go  to  the  post  office,  then  on  to  his 
cousin's  house,  and  home  again  ?     How  many  rods  ? 

83.  ^lake  a  table  showing  the  12ths  of  72  up  to  -^|. 

84.  Express  the  ratio  of  each  of  the  first  12  multiples 
of  6  to  72. 

85.  A  field  is  9  rd.  long  and  8  rd.  wide.  How  many 
so     rd     in   -i.   of    if^     -5-'^     -^"^     il'^     ^-'^     1'^     ^'^     l*^ 

&q.    iu.     ill     ^2     ^^     ^^  '        12"        12*        12"        2*        4*       ?*        3* 

1'?      19       5  ? 
3  •        6  •        6  • 

86.  What  part  of  a  field  12  rd.  long  and  6  rd.  wide  is 
a  lot  2  rd.  long,  and  2  rd.  wide?  4  rd.  long  and  3  rd. 
wide?  9  rd.  long  and  2  rd.  wide?  8  rd.  long  and  3  rd. 
wide?  6  rd.  square?  11  rd.  long  and  6  rd.  wide?  12 
rd.  long  and  4  rd.  wide?     15  rd.  long  and  4  rd.  wide  ? 

87.  Draw  a  diagram  of  a  right  triangle  whose  base  is 
6  rd.  and  altitude  3  rd.,  drawing  to  a  scale  of  1  inch  to 
a  rod.     Find  its  area. 

88.  What  is  the  ratio  of  that  triangle  to  a  rectangle 
6  rd.  long  and  J  as  wide  as  long  ?     Draw. 

89.  HoAv  many  inch  cubes  will  be  needed  to  make  a 
square  prism  6  in.  long,  3  in.  wide,  and  2  in.  high  ? 

90.  How  many  cu.  ft.  in  a  tank  6  ft.  long,  5  ft.  wide, 
and  7  ft.  high? 

91.  How  many  cu.  iuo  of  air  can  there  be  in  a  box  6  in. 
long,  4  in.  wide,  and  3  in.  high  ? 

92.  If  a  solid  3  in.  long,  2  in.  wide,  and  2  in.  high  were 
placed  in  the  box  mentioned  in  Ex.  91,  how  many  cu. 
in.  of  air  would  be  left  in  it  ? 


188  SIXES 

93.  How  many  cu.  ft.  of  air  can  there  be  in  a  room 
60  ft.  long,  40  ft.  wide,  and  12  ft.  high? 

94.  If  a  solid  6  ft.  long,  5  ft.  wide,  and  4  ft.  high 
were  placed  in  tlie  room  mentioned  in  Ex.  93,  how  many 
cu.  ft.  of  air  would  be  left  in  the  room? 

95.  Estimate  the  length,  width,  and  height  of  the 
schoolroom,  and  find  about  how  many  cu.  ft.  of  air  the 
room  will  hold. 

Let  the  children  suggest  different  rectangular  solids  to  be  placed 
in  the  room,  and  find  the  cu.  ft.  of  air  displaced  by  them. 

96.  What  is  meant  by  ^  of  anything? 

97.  Draw  a  picture  of  a  pie  cut  into  sixths,  and  tell 
how  many  sixths  of  a  pie  in  2  equal  pies.  In  4.  7. 
8.     5. 

98.  How  many  sixths  in  1}  ?    In  2^  ?    In  3f  ?    In  91  ? 

In  8|  ?     In  71  ?     In  6|  ? 

^^*       6~'  6~"  6~~'  6~'  6 6 

100.  Co]3y  Fig.  1  by  placing  equilteral 
triangles.  How  many  sides  has  the  figure  ? 
A  figure  bounded  by  six  straight  lines  is 
called  a  Hexagon. 

101.  How  long  would  the  perimeter  of 
the  hexagon  be  if  each  side  of  a  triangle  Avere  6  in.  long  ? 
8  in.?     9  in.?     7  in.? 

102.  How  many  such  hexagons  could  you  make  from 
42  such  triangles  ?     36  ?     72  ?     24  ?     54  ?     66  ? 

103.  Divide  your  hexagon  into  trapezoids.     What  is 
the  ratio  of  each  trapezoid  to  the  hexagon  ? 

104.  Divide  your  hexagon  into    rhombuses.     What  is 
the  ratio  of  each  rhombus  to  the  hexagon  ? 


T^IG.  1 


SIXES  189 

105.  One  of  the  triangles  is  what  part  of  the  hexagon 
you  have  made  ?  3  triangles  ?  2  triangles  ?  4  triangles  ? 
5  triangles  ? 

106.  Divide  each  triangle  that  makes  your 
hexagon  into  2  right  triangles,  as  in  Fig.  2. 
Hold  up  one  of  the  right  triangles,  and  show 
its  right  angle. 

107.  Each  of  your  right  triangles  is  what         -^^^-  ^ 
part  of  an  equilateral  triangle  ?     What  part  of  the  hexa- 
gon ?     J  of  1  =  ? 

108.  Draw  several  hexagons  of  different  shapes. 

109.  Form  6  equilateral  triangles  into  a  figure  that  is 
not  a  hexagon.     Show  J  of  J  of  the  figure. 

110.  What  kind  of  a  fraction  is  i  of  |  ?  How  can  the 
value  of  that  kind  of  fractions  be  found  ? 

111.  Find  value  : 

iofj        Joff        ^ofi        fof^        iofl       fofl 

foff  foff  foff  fofl  fofl  I  of  A 

lofi         foff         foff         I  of  I       JjOfl     ^Vofi 

5     nf  1  5     nf  A         JL.  nf  1         _9_  of  5  _8_  of  ^       -9-  of  ^ 

11  ^^6  TT  '^^  6  12   ^^  6  12   ^^   6  12   ^^  6         12   ^^  6 

112.  How  many  rods  in  1  mile  and  80  rods  ? 

113.  George  rode  on  his  bicycle  1  mile  lacking  10  rods. 
How  many  rods  did  he  ride  ? 

114.  At  6  cents  a  pt.,  what  will  be  the  cost  of  3  qt.  of 
milk  ?     Of  a  gallon  ?     Of  a  gallon  and  a  half  gallon  ? 

115.  At  6  cents  a  qt.,  how  much  will  a  pk.  of  beans 
cost?     A  pk.  and  a  half?     A  bu.? 

116.  Mary  borrowed  60  cents  from  her  brother,  and 
paid  him  6  cents  of  it  every  week,  How  many  weeks  did 
it  take  her  to  pay  the  whole  ? 


190  SIXES 

117.  Paying  6  cents  a  week,  how  many  weeks  would  it 
have  taken  her  to  pay  48  cents  ?  72  cents  ?  78  cents  ? 
84  cents  ? 

118.  Sometimes  when  people  borrow  money  they  pay  6 

cents  for  every  dollar  that  they  keep  for  a  whole  year, 

besides  paying  back  the  dollar.     If  you  lent  i  3  for  a  year 

at  that  rate,  how  much  would  you  receive  at  the  end  of 

the  year  besides  the  $  3  you  had  lent  ? 

Explain  that  interest  is  paid  for  the  use  of  money  just  as  horse 
hire  is  paid  for  the  use  of  a  horse,  or  rent  for  the  use  of  a  house. 

119.  The  money  which  is  paid  for  the  use  of  money  is 
called  Interest.  If  6  cents  are  paid  for  the  use  of  a  dol- 
lar for  one  year,  how  much  interest  must  be  paid  if  it  is 
kept  4  years  ?     7  years  ?     9  years  ?     6  years  ?     8  years  ? 

120.  If  the  interest  of  a  dollar  for  one  3^ear  is  6  cents, 
what  is  the  interest  of  3  dollars  for  a  year  ?    Of  i  5  ?    f  7  ? 

19?  112?  16?  18?  14?  .$11? 

121.  John  lent  $S  for  a  year  at  6  per  cent.  (That 
means  6  cents  for  every  dollar.)  How  much  interest  did 
he  get  ? 

122.  At  6  per  cent,  what  is  the  interest  of  -f  1  for  IJ 
years  ?     For  2 J  years  ?     4J  years  ?     6^  years  ?     8^  years  ? 

123.  James  lent  some  money  at  6  per  cent,  and  received 
42  cents  interest  at  the  end  of  a  year.  How  many  dollars 
did  he  lend  ?  How  many  dollars  must  he  lend  at  that 
rate  in  order  to  get  72  cents  ?     36  ?     54  ?     66?     48  ? 

124.  Arthur  has  a  dollar  in  a  bank  that  pays  6  per  cent. 
How  much  interest  will  it  give  him  in  2^  years  ?  In  4J 
years  ?     In  5^  years  ?     In  9i  years  ? 

125.  What  is  the  interest  of  a  dollar  for  7  years  at  5 
percent?  (5  cents  for  every  dollar.)  For  9  years?  P^or 
12  years  ? 


SIXES  191 

126.  How  much  will  William  receive  at  the  end  of  a 
year  on  each  dollar  that  he  has  in  a  bank  which  pays  3 
per  cent  ?     How  much  would  he  receive  for  i  10  ? 

127.  If  a  bank  paid  5  per  cent,  how  much  would  May 
receive  at  the  end  of  a  year  if  she  had  $  8  in  it  ?  How 
much  if  she  had  1 5  ?     17?     14?     |10?     |9? 

128.  At  6  per  cent,  how  long  must  I  keep  a  borrowed 
dollar  to  pay  2-1  cents  interest  ?  To  pay  18  cents  ?  54: 
cents  ?    72  cents  ?    60  cents  ? 

129.  Which  gives  the  more  interest  at  the  end  of  a 
year :  $  8  loaned  at  6  per  cent,  or  f  7  at  7  per  cent  ? 

130.  Per  cent  is  sometimes  written  %.  What  is  the 
interest  of  a  dollar  for  3  years  at  6  %  ?    For  3  years  at  8  %  ? 

131.  Anna  may  tell  how  much  money  she  would  like 
to  have  at  interest,  and  how  much  it  would  bring  her 
each  year  at  6%.     At  5%.     At  4%. 

Let  this  exercise  be  general. 

132.  If  it  costs  §286.50  to  make  6  wagons,  how  much 
will  it  cost  to  make  1  wagon  ?     5  wagons  ? 

133.  If  it  costs  $47.75  to  make  one  wagon,  for  how 
much  must  the  maker  sell  it  to  gain  -f  5.25? 

134.  At  6  cents  a  ft.,  how  much  will  it  cost  to  fence 
a  lot  1  rd.  square  ?     2  rd.  square  ?     20  rd.  square  ? 

Notice  that  in  the  following  examples  one  unit  of  the  minuend 
must  be  reduced  to  fractional  units. 

135.  From        27f        38  47  56  28  96 

take        lli       Ji        ii       Ji        ^       _1 

136.  Write  in  Roman  notation  two  factors  of  77. 


CHAPTER   XV 

TWELVES 

Long  Division,  Square  Foot,  Cubic  Foot,  Common 

Multiple 


NUMBER 

TABLE 

1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

m 

66 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

0 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1.  Count  by  twelves  to  96.     How  many  twelves  did 
you  count? 

2.  Count   by   twelves   from    96   to    144.      How   many 
twelves  from  96  to  144? 

3.  Count  backwards  by  twelves  from  144  to  0. 

4.  Write  and  learn  the  table  of  twelves  up  to  144. 

192 


TWELVES  193 

5.  How  many  twelves  in  48  ?     72  ?     36  ?     96  ?     60  ? 

6.  Name  the  7th  multiple  of  12  ;  the  5th  ;   9th  i  6th. 

7.  Add  2  twelves  to  24,  60,  36,  84,  12,  48,  72,  108,  96. 

8.  Take  2  twelves  from  96,  60, 144,  108,  48,  84,  36,  72. 

9.  How  many  twelves  must  he  added  to  24  to  make 
60?     36?     72?     48?     96? 

10.  How  many  twelves  must  be  subtracted  from  96  to 
leave  72  ?     48  ?  "^  24  ?     84  ?     36  ?     60  ? 

11.  How  many  twelves  must  be  added  to  36  to  equal  6 
twelves  ?    5  twelves  ?   9  twelves  ?    7  twelves  ?    12  twelves  ? 

12.  How  many  twelves  must  be  taken  from  108  to  leave 
7  twelves  ?   4  twelves  ?    6  twelves  ?    3  twelves  ?   5  twelves  ? 

13.  3  twelves  +  5  =  ?  7  twelves  +  9  =  ?  9  twelves +  10  =  ? 

14.  9  twelves  — 4  =  ?  7  twelves  — 6  =  ?  4  twelves  — 9  =  ? 

15.  How  many  sixes   in   2  twelves?      3  twelves?      5 
twelves  ?     8  twelves  ?     6  twelves  ? 

16.  How  many  threes  in  12?  In  2  twelves?  3  twelves? 
4  twelves  ?     6  twelves  ?     5  twelves? 

17.  How  many  fours  in  12  ?    In  2  twelves  ?    3  twelves  ? 
4  twelves  ?     5  twelves  ?     7  twelves  ? 

18.  How  many  twelves  in  4  sixes  ?  In  10  sixes  ?  8  sixes  ? 

19.  How  many  twelves  in  3  eights  ?  In  6  tens  ?  12  sixes  ? 

20.  12  is  ^  of  what  number  ?     i  of  what?     J  of  what  ? 
1  of  what  ?     -^  of  what  ?     -i  of  what  ?     -l  of  wliat  ? 

21.  Make  a  list  of  the  multiples  of  12  that  are  less  than 
145  and  of  the  factors  that  compose  them,  as : 

12  =  2  X  6  or  3  X  4 

24  =  2  X  12  or  3  X  8  or  4  X  6. 

■  22.    There  are  four   numbers  besides  12   that  are  con- 
tained in  every  multiple  of  12.     Name  them. 

HORN.  ARITH. 13 


194  TWELVES 

23.  Add  the  number  whose  factors  are  8  and  7  to  the 
number  whose  factors  are  8  and  9,  How  many  eights 
does  their  sum  contain  ? 

24.  Find  the  difference  between  the  number  that  is 
composed  of  8  and  6  and  the  number  composed  of  4  and  12. 

25.  What  is  the  ratio  of  the  number  whose  factors  are 
3  and  7  to  the  number  whose  factors  are  5  and  7  ? 

26.  How  many  hours  does  it  take  the  hour  hand  of  a 
clock  or  watch  to  go  once  around  the  face  ?  How  many 
hours  to  go  one  half  the  way  around  ?  -J  of  the  way  ? 
■|  of  the  way  ?     i  of  the  way  ?     -I  of  the  way  ? 

27.  How  many  hours  does  it  take  for  the  hour  hand  to 
go  around  twice  ?     3  times  ?     5  times  ?     6  times  ? 

28.  In  24  hours,  how  many  times  does  the  hour  hand  go 
around  the  face  of  a  clock  ?     In  60  hr.  ?     72  hr.  ?     96  hr.  ? 

29.  How  many  times  does  the  minute  hand  of  a  clock 
go  around  the  face  between  noon  and  midnight?  How 
many  times  between  noon  to-day  and  noon  to-morrow  ? 
Between  noon  on  Monday  and  noon  on  Wednesday  ? 

30.  Name  the  months  of  the  year  in  order,  beginning 
with  January. 

31.  How  many  months  in  2  years  ?  In  4  years  ?  In  8 
years  ?     In  10  years  ?     In  12  years  ?     In  ^^  of  a  year  ? 

32.  How  many  years  in  25  months  ?  39  months  ?  50 
months  ?     67  months  ?     109  months  ?     88  months  ? 

33.  If  you  save  a  dollar  a  month  for  3  years,  how  much 
money  will  you  save  ? 

34.  HoAv  much  money  will  you  save  if  you  lay  aside  one 
dollar  a  month  for  2J  years  ?     For  5  years  ?     3 J  years  ? 

35.  How  many  months  had  you  lived  when  you  were 
just  6  years  old  ? 


TWELVES  195 

36.  Margaret  was  9  years  old  on  the  1st  day  of  last 

month,     llow  many  months  has  she  lived  and  how  many 

days  over? 

Let  each  child  in  the  class  reckon  up  the  number  of  whole  months 
he  has  lived  and  the  days  over. 

37.  How  many  eggs  in  7  dozen  ?  12  dozen  ?  3J  dozen  ? 
8^  dozen  ?     2i  dozen  ?     6 J  dozen  ? 

38.  How  many  dozen  eggs  in  48  eggs?     72  eggs? 

39.  Gertrude  is  visiting  her  cousin  Alice.  She  has 
spent  12  days  at  Alice's  house  and  -i  of  her  visit  is  gone. 
How  long  was  the  visit  to  be  and  how  many  days  longer 
can  she  stay  ? 

40.  How  many  in.  long  is  a  line  that  is  2  ft.  long? 
2i  ft.?     31  ft.?     41  ft.?     6  ft.?     71  ft.?     81  ft.?     9  ft.? 

41.  How  many  ft.  long  is  a  line  that  is  86  in.?  60  in.? 
84  in.?     108  in.? 

42.  Draw  on  the  board  a  square  1  foot  each  way,  and 
divide  it  off  into  square  inches.  How  many  rows  of 
squares?  How  many  squares  in  eacli  row?  How  many 
squares  in  all  ? 

43.  How  many  sq.  in.  in  2  sq.  ft.?     3  sq.  ft.?     5  sq.  ft.? 

44.  How  many  sq.  in.  in  a  square  which  is  2  ft.  long? 

45.  How  many  sq.  in.  in  a  sq.  yd.? 

The  square  foot  with  its  subdivisions  of  square  inches  should 
remain  on  the  board  where  the  children  can  see  it  for  several  davs, 
and  occasional  short  drills  should  be  given  by  questions  similar  to 
those  relating  to  dozens. 

46.  What  is  the  ratio  of  12  to  24  ?     12  to  36  ? 
Use  chart  drill. 

47.  How  many  inches  in  i  of  a  yd.?     In  J  of  a  yd.? 

48.  What  is  the  ratio  of  12  to  48?  What  is  |  or  -J  of 
48?     I  of  48? 


196  TWELVES 

49.  A  lot  has  a  frontage  of  12  rods  and  a  depth  of 
48  rods.  What  is  the  ratio  of  the  length  of  the  front 
fence  to  the  side  fence  ?  How  many  rods  of  fencing  will 
it  take  to  inclose  it?  How  much  will  the  fence  cost  at 
1.15  a  ft.? 

50.  What  is  the  ratio  of  12  to  60  ?  What  number  is 
I  of  60  ?     I  of  60  ?     f  of  60  ? 

51.  A's  house  is  60  rd.  from  B's  house.  When  A  has 
gone  |-  of  the  Avay  to  B's  house,  how  many  rd.  is  he  from 
his  own  house  ?  How  far  is  he  from  B's  house  ?  How 
far  is  he  from  each  house  when  he  has  gone  ^  of  the  way 
to  B's  house  ? 

52.  Susan  has  to  practice  on  the  piano  an  hour  every 
morning.  How  many  fifths  of  her  practice  hour  are  past 
when  she  has  practiced  24  minutes  ?     48  minutes  ? 

53.  Fill  out  and  learn  : 

i  of  72  =  The  ratio  of  — to  72  is  — 

f  or  1  of  72  =  The  ratio  of —  to  72  is  — 

f  or  ^  of  72  =  The  ratio  of —  to  72  is  — 

I  or  f  of  72  =  The  ratio  of —  to  72  is  — 

I-  of  72  =  The  ratio  of  — to  72  is  — 

b 

54.  A  farmer  buys  a  wagon  for  $  72  and  pays  $  12  cash. 
What  fractional  part  of  the  jjrice  does  he  pay,  and  what 
part  does  he  still  owe  ? 

55.  At  the  end  of  six  months  he  pays  1 24  more.  What 
part  of  the  price  does  he  still  owe  ? 

56.  He  pays  1 24  at  the  end  of  another  six  months. 
What  part  of  the  price  does  he  still  owe  ? 

57.  Draw  a  rectangle  1  ft.  long  and  6  in.  wide.  Divide 
it  into  6ths.  How  many  sq.  in.  in  each  sixth?  How 
many  in  |  of  it  ?     In  |  or  i  of  it  ?     In  |  or  |  of  it  ? 


TWELVES  197 

58.  Make  a  table  showing  sevenths  of  84  from  ^  to  ^. 

59.  Make  a  table  showing  the  ratio  to  84  of  each  multi- 
ple of  12  less  than  100. 

60.  A  fence  84  ft.  long  costs  $21.  12  ft.  of  the  fence 
cost  what  part  of  the  money  ?  How  many  dollars  ?  How 
many  dollars  do  36  ft.  cost?     60  ft.?     72  ft.?     24  ft.? 

61.  Make  a  table  of  the  8ths  of  96  as  far  as  |. 

62.  IMake  a  table  showing  the  ratio  of  each  of  the  first 
8  multiples  of  12  to  96. 

63.  If  an  acre  of  land  costs  %  96,  how  much  do  |  of  an 
acre  cost?     f?     |?     |?     i?     |? 

64.  160  square   rods  make  an  acre.      Into  how  many 

pieces  of  ground  2  rods  square  can  an  acre  be  divided  ? 

Let  children  get  ideas  of  an  acre  by  measuring  off  distances  on  the 
playground,  estimating  the  length  of  a  line  about  290  feet  long  and 
the  area  of  a  square  of  these  dimensions. 

65.  A  farmer  has  a  farm  of  96  acres.  12  acres  are 
planted  with  potatoes,  24  with  corn,  36  with  wheat,  and 
the  rest  is  pasture  land.  What  part  of  the  farm  is  planted 
with  potatoes?  Corn?  Wheat?  What  part  is  pasture 
land? 

66.  Make  problems  about  8ths  of  96. 

67.  Make  a  table  showing  9ths  of  108  from  J  to  |-. 
Make  a  table  showing  the  ratio  of  each  of  the  first  9  mul- 
tiples of  12  to  108. 

68.  The  cost  of  12  pencils  will  be  what  part  of  the  cost 

of  108  pencils  ?     Supposing  that  108  pencils  cost  45  cents, 

how  much  would  12  pencils  cost  ? 

Let  children  suppose  different  prices  for  different  numbers  of 
pencils. 

69.  What  part  of  the  cost  of  108  pencils  is  the  cost  of 
24  pencils  ?     Of  36  pencils  ?     60  pencils  ?     72  pencils  ? 


IC)^  TWELVES 

70.  If  1  dozen  pencils  cost  5  cents,  how  many  times  as 
mncli  will  108  pencils  cost  ?  How  many  cents  would  that 
be? 

71.  Make  problems  about  9ths  of  108. 

72.  Make  a  table  of  the  lOths  of  120  up  to  1^. 

73.  Make  a  table  showing  the  ratio  to  120  of  each  of  the 
multiples  of  12  that  are  less  than  120. 

74.  John  had  il.20  and  spent  12  cents.  What  part  of 
his  money  did  he  spend  ?     What  part  had  he  left  ? 

75.  What  part  had  he  spent  and  what  part  was  left 
when  he  had  spent  24  cents?     48  cents?     1.60?     |.72? 

76.  Make  a  table  showing  yL  to  i|  of  132. 

77.  Make  a  table  showing  the  ratio  to  132  of  each  of 
the  multiples  of  12  less  than  132. 

78.  How  long  is  a  line  that  is  -f-^  as  long  as  a  line  132 
inches  long  ?     -/y  '^^  ^^^^^  •     TT  ^^  ^^^^^  ^      ri  ^     t\  ^     1 1  ■ 

6    '/>       12  9       _4_  V       _8^  9       11  9 
TT-       11-       11-       11-       11- 

79.  A  certain  house  is  132  miles  from  Philadelphia. 
What  fraction  of  that  distance  has  a  man  traveled  who 
has  gone  12  miles  toward  Philadelphia  ?  What  part  of 
the  distance  does  he  still  have  to  travel? 

80.  When  he  has  traveled  24  mi.,  what  part  of  the  dis- 
tance has  he  traveled  and  what  part  remains  ?  What  part 
has  he  gone  over,  and  what  part  remains,  when  he  has 
traveled  48  mi.  ?     60  mi.  ?     84  mi.  ?     96  mi.  ?     120  mi.  ? 

81.  Make  a  table  showing  12ths  of  144  from  -f^  to  ||. 

82.  Make  a  table  showing  the  ratio  of  each  of  the  first 
12  multiples  of  12  to  the  number  144. 

83.  A  jar  of  butter  is  priced  at  11.44.  How  much  will 
-JL  of  it  cost  *?     -3^  or  4  of  it  '^     -5-  '^     -"-  '^     -^-  '^     -9-  '^     ^1  *? 

12   ^^  ^^  ^^^^  '        12^  Ol   ^  Oi  lU  .       -j^2  •        12  •        12  •        12  •       12  • 


TWELVES  199 

84.  What  part  of  a  square  foot  is  48  square  inches  ?  24 
sq.  in.  ?     60  sq.  in.  ?     96  sq.  in.  ?     120  sq.  in.  ? 

85.  How  many  inches  of  ribbon  will  it  take  to  bind  a 
lamp  mat  that  is  one  foot  square  ? 

86.  What  is  the  square  of  12  ? 

87.  -J  of  Harry's  money  is  12  cents.  How  much  has  he  ? 
How  many  cents  had  he  when  ^  of  his  money  was  9  cents  ? 
7  cents  ?     11  cents  ?     13  cents  ? 

88.  Louisa  has  12  cents.  John  has  3  times  as  many 
cents  and  5  cents  more.      How  many  cents  has  John  ? 

89.  If  i\lary  had  3  cents  more,  she  would  have  twice 
as  much  as  Jennie,  who  has  12  cents.  How  many  cents 
has  Mary  ? 

90.  Divide  276  by  12  by  long  division. 

SOLUTION 

12)276(23 

i)A  Show  the  process  and   let  the  children  become 

"■^  expert  in  it  before  giving  an  explanation  of  it. 

36 

91.  By  long  division  find  the  quotient  of  288  and  12. 

92.  By  12  divide  1728,  3456,  432,  264,  384,  636. 

93.  Divide  373  by  12  and  see  if  you  get  the  answer 

94.  Divide  by  12  each  of  the  numbers  between  500 
and  600  whose  unit  figure  is  6. 

95.  Divide  by  11  each  of  the  numbers  between  600 
and  700  whose  unit  figure  is  5.  ' 

96.  How"  many  feet  in  189  inches  ?     474  in.?     699  in.? 

97.  How  many  dozen  eggs  in  972  eggs  ?      852  eggs  ? 


200 


TWELVES 


98.  John  was  given  #2.64  with  which  to  buy  coffee. 
How  many  lb.  could  he  buy  at  12  cents  a  lb.?  At  11 
cents  a  lb.? 

99.  How  many  years  in  898  months  ?  961  months  ? 
6846  months  ?     7849  months  ? 

100.  At  his  last  birthday  Mr.  Smith  had  lived  336 
months.     How  many  years  old  was  he  ? 

101.  How  many  years  old  will  you  be  when  you  have 
lived  216  months?  348  months?  396  months?  456 
months  ? 

102.  A  dozen  readers  cost  14.20.  What  is  the  price 
of  each  ? 

103.  How  much  rent  does  Mr.  Jones  pay  each  month 
when  his  yearly  rent  is  $  288  ?  How  much  when  his 
yearly  rent  is  -f  384  ? 

104.  William  has  12.00  at  interest  at  6%.  How  much 
does  it  gain  each  year  ?     In  how  many  years  Avill  it  gain 

11.56?     12.16?     $3.72? 

105.  Fig.  1  represents  a 
floor  12  ft.  long  and  6  feet 
wide.  How  many  square  ft. 
Q  are  represented  by  AB  CD  ? 
How  many  square  yd.  will  it 
take  to  cover  the  floor  ?  Draw 
the  figure  and  outline  the 
square  yd. 


D 

B 


Fig.  1 


106.  Draw  a  figure  to  represent  a  square  floor  12  ft. 
long,  and  show  how  many  square  yd.  of  linoleum  it  will 
take  to  cover  it  by  outlining  with  a  heavy  line  each  figure 
that  represents  a  scpuire  yard. 


TWELVES 


201 


Fig.  3 


107.  Arrange  equilateral  triangles  as  in 
Fig.  2.  Show  ^  of  the  figure  and  tell  how 
many  12ths  it  equals. 

108.  Separate  the  figure  into  4  trapezoids. 
^  =  how  many  12ths  ?    |^  =  how  many  12ths  ? 

109.  Separate  the  figure  into  6  rhombuses. 
1  =  how  many  12ths  ?     |  =  how  many  12ths  ? 

110.  How  long  is  the  perimeter  of  Fig.  2, 
if  each  side  of  the  triangles  is  12  in.  long  ?    9  in.  ?    8  in.? 

111.  If  the  perimeter  of  Fig.  2  is  70  inches,  how  long 
is  each  side  of  the  triangles  ? 

112.  How  many  triangles  in  this  six- 
pointed  star  ?  How  many  triangles 
would  it  take  to  make  9  such  stars  ?  6 
such  stars  ?    11  such  stars  ? 

113.  Copy  Fig.  3  by  placing  equilat- 
eral triangles.  Separate  your  copy  into 
6  equal  rhombuses.  Each  rhombus  is 
what  fractional  part  of  the  figure  ?    Each 

triangle  is  what  fractional  part  of  a  rhombus  ?      J  of  J  =  ? 

114.  Remove  triangles  from  the  figure  until  you  have 
a  hexagon  left.  What  fractional  part  of  the  figure  did 
you  take  away  ? 

115.  Copy  Fig.  4  by  placing 
equilateral  triangles.  Separate  your 
figure  into  4  equal  trapezoids. 
Each  trapezoid  is  what  part  of  the 
whole  figure  ?  Each  triangle  is  what 
part  of  a  trapezoid  ?     -i  of  |^  =  ? 

116.  If  the  perimeter  of  Fig.  4  were  396  inches,  how 
lonof  would  each  side  of  the  triano^les  be  V 


Fig.  3 


Fig.  4 


202 


TWELVES 


¥m. 


117.  Can  you  make  Fig.  5  by 
changing  the  place  of  one  of  the 
trapezoids  in  Fig.  4  ? 

118.  Show  by  Fig.  5  that  ^  is 
equal  to  -J  of  i. 

119.  How  long  would  each  side 
of  the  triangles  be  if  the  perimeter  of  the  figure  were 
72  in.  ?     396  in.  ?     528  in.  ? 

120.  How  many  inch -cubes  would  it  take  to  make  a 
layer  of  them  12  in.  long  and  12  in.  wide  ? 

121.  How  many  inch -cubes  would  it  take  to  make  2 
such  layers  ?  ,  8  layers  ?    9  layers  ?    11  layers  ?    12  layers? 

122.  A  cubic  foot  is  12  inches  long,  12  inches  wide,  and 

12  inches  liigh.     How  many  cubic  inches  make  a  cubic 

foot  ? 

Let  children  show  their  ideas  of  the  dimensions  of  a  cu})ic  foot  by 
outlining  with  their  hands. 

123.  Find  how  many  cu.  in.  make  2  cu.  ft.     3  cu.  ft. 
5  cu.  ft.      7  cu.  ft.      9  cu.  ft.      11  cu.  ft. 

124.  How  many  cubic  inches  in  1  of  a  cubic  foot  ? 
2^  cu.  ft.?     S^  cu.  ft.?     41  cu.  ft.?     6^  cu.  ft.? 

125.  A  line  1  in.  long  is  what  fractional  part  of  a  line 
1  ft.  long  ?    Half  an  inch  is  what  part  of  a  foot  ?   J  of  i2-  =  ? 

126.  AVhat  kind  of  a  fraction  is  2  ^^  i^  ^     How  do  you 
find  the  value  of  it  ? 


127. 
128. 

129. 

±S.  —  9 
12   "■  • 


3   uj-   ^  2         "        3  ^^   1 2         °        5   ^^   1 2 


6    of  -6_.  _  ? 

y  ^^  12  — ' 


A  square  inch  is  what  part  of  a  square  foot  ? 

4A  =  ?   4  8  _  ?   14  =  ?  1X3.  =  9 
12    •    1^   •    12    • 


48  _? 
1^ 

6  5.—?   1#  —  ? 
12  ""•    12  ~~  • 


12 


1X3. 
12 

^#  = 
12 


1^  =? 

12 

12   • 


13  —? 
12 

JL4^  =  9 
12 


130.    At  12  cents  a  yd.,  how  much  ribbon  can  be  bought 
for  29  cents?     f.31?     |.89?     11.29?     *2.89?     #1.58? 


TWELVES  203 

131.  Name  the  unit  figure  of  each  of  the  multiples  of 
12  given  in  their  order  up  to  144.  Can  any  of  these  mul- 
tiples of  12  be  odd  numbers  ? 

132.  Which  multiple  of  12  is  36  ?  How  many  times 
does  36  contain  12  ? 

133.  Think  of  a  number  and  of  one  of  its  multi})les,  and 
see  if  this  definition  is  true.  "  A  multiple  of  a  number  is 
a  number  that  will  contain  it  Avithout  a  remainder." 

134.  There  is  one  multiple  of  12  less  than  100  that  is 
also  a  multiple  of  5.     What  is  it  ? 

135.  What  number  less  than  100  is  a  multiple  of  12 
and  also  of  7  ? 

136.  When  a  number  is  a  multiple  of  two  other  num- 
bers, it  is  called  a  Common  Multiple  of  them.  Name  a 
common  multiple   of  5   and   12,    8   and   7,    11   and  5. 

137.  42  is  a  common  multiple  of  6  and  what  other 
number  ? 

138.  Turn  to  the  number  table  of  9's,  and  see  how  many 
of  the  multiples  of  9  in  it  are  also  multiples  of  12.  Make 
a  list  of  these  common  multiples  of  9  and  12.  Which  is 
the  least  ? 

139.  Make  a  list  of  the  multiples  of  12  and  of  8  that 
you  have  learned.  Show  those  that  are  common  multiples 
of  12  and  8.     Which  is  the  least  common  multiple  ? 

140.  Turn  to  the  number  table  of  6's,  and  the  number 
table  of  8's,  and  you  will  see  that  the  multiples  of  6  meet 
the  multiples  of  8  in  the  numbers  24,  48,  72,  96.  What 
is  the  least  number  that  contains  both  8  and  6  ? 

A  way  of  showing  this  meeting  of  the  multiples  is  to  have  tha 
pupils  put  the  first  hundred  numbers  on  the  board,  writing  all  the 
multiples  of  one  number  with  crayon  of  a  certain  color  and  those  of 
other  numbers  with  other  colors.     For  instance,  if  the  multiples  of  6 


204  TWELVES 

are  red,  8  blue,  and  9  yellow,  72  will  show  itself  in  its  varicolored 
representation  as  a  multiple  of  them  all. 

141.  Make  a  list  of  the  common  multiples  of  6  and  9 
tliat  are  less  than  100,  and  tell  which  is  the  least  common 
multiple. 

142.  What  is  the  least  common  multiple  of  6  and  7  ? 
6  and  11  ?     6  and  5  ? 

143.  Find  the  least  common  multiple  of  6  and  8  without 
looking  at  the  number  tables. 

Show  the  following  method :  To  find  the  least  common  multiple 
of  two  numbers,  take  the  largest  of  the  numbers  and  name  its  multi- 
ples in  order  until  one  is  found  that  is  also  a  multiple  of  the  other 
number.  Thus,  to  find  the  least  common  multiple  of  6  and  8,  name 
the  multiples  of  8  in  order,  8,  16,  24,  trying  each  one  to  see  if  it  will 
contain  6,  until  one  is  found  that  will  contain  it. 

144.  Find  the  first  number  in  which  the  multiples  of  8 
meet  the  multiples  of  3.  In  what  number  do  the  multiples 
of  8  first  meet  the  multiples  of  5  ?     7  ?     10  ? 

145.  Find  the  least  common  multiple  of  12  and  9,  12 
and  8,  12  and  10,  12  and  7. 

146.  Find  the  least  common  multiple  of  9  and  6,  9  and 
4,  9  and  5,  9  and  8,  8  and  10,  10  and  6. 

147.  Mary  may  name  two  numbers  less  than  13,  and 
the  class  may  find  the  least  common  multiple  of  them. 

Let  this  exercise  be  general. 

148.  6  is  a  common  multiple  of  what  two  other  numbers? 

149.  Of  what  two  numbers  is  35  the  least  common  mul- 
tiple?    21?     77?     15? 

150.  Write  in  Roman  notation  the  least  common  mul- 
tiple of  12  and  7. 


CHAPTER  XVI 
REVIEW 

Average,    Commox    Divisor,    Adding,    Subtracting,    and 
Multiplying  Denominate  Numbers,  Per  Cent,  Bills 

1.  Find  by  long  division  the  q^uotient  of  748  and  11. 
Of  799  and  12. 

2.  Fill  out  this  table  of  the  products  of  21  multiplied 

by  numbers  from  2  to  9. 

application 

21  X  2=42  21)1491(71 

21  X  3  =  63  147_ 

21  X  4  =  21 

21 

3.  Divide  2541  by  21. 

Tables  of  products  are  very  helpful  to  pupils  in  beginning  long 
division,  but  they  should  be  encouraged  to  dispense  with  them  as  soon 
as  possible,  and  to  estimate  their  trial  quotients  carefully. 

4.  Find  how  many  times  21  is  contained  in  each  of  the 
following  numbers ;  2352,  2583,  2982,  3003,  2835. 

5.  By  31  divide :  3503,  3787,  3875,  4154,  3906,  4092. 

6.  By  22  divide :  2706,  2662,  2816,  2984,  2882,  2728. 

7.  How  many  times  is  32  contained  in  3872  ?     4224  ? 

8.  By  41  divide  :  5125,  5882,  4961,  5371,  5494,  5412. 

9.  Use  42  as  a  divisor  of :  5337,  5166,  5418,  5124,  5712. 

10.  How  many  times  is  25  contained  in  1775  ?     8126  ? 

11.  How  many  times  does  879  contain  22  ?     32  ?     62  ? 

12.  How  many  times  does  the  square  of  21  contain  49  ? 


206  REVIEW 

13.  If  31  men  own  in  equal  shares  a  mine  whicli  pays 
118,775  this  year,  what  is  each  man's  share  of  the  profits  ? 

14.  How  much  is  /^  oi  6824  ?     8965  ?     4869  ?     12428  ? 

15.  If  16  lb.  of  beef  cost  $2.56,  how  much  does  1  lb. 
cost  ? 

16.  At  i  .33  a  gallon,  how  many  gallons  of  vinegar  can 
be  bought  for  14.62?    |4.95?    17.92?    $8.91?    $8.25? 

17.  Think  of  a  number  that  is  as  much  greater  than  10 
as  it  is  less  than  14. 

18.  Find  a  number  that  is  as  much  less  than  20  as  it  is 
greater  than  12. 

19.  What  number  is  halfway  between  10  and  20?  24 
and  30  ?     40  and  50  ?     25  and  45  ? 

20.  A  number  that  is  halfway  between  two  numbers  is 
called  the  average  of  those  numbers.  What  is  the  aver- 
age of  16  and  20  ?     18  and  22  ? 

21.  How  long  is  a  line  whose  lengtli  is  the  average  of 
an  8-inch  line  and  a  10-inch  line  ? 

22.  To  find  the  average  of  two  numbers,  divide  their 
sum  by  2.  Find  the  averages  in  the  last  five  examples  in 
that  way. 

23.  Julia  is  12  years  old  and  Jennie  is  16  years  old. 
What  is  the  average  of  their  ages  ? 

24.  What  is  the  average  of  146  and  178  ?     234  and  478  ? 

25.  Sometimes  the  average  of  numbers  is  a  fractional 
number.  Find  the  average  of  7  and  8,  17  and  20,  46 
and  53,  18  and  47. 

26.  Find  the  average  of  19  and  49,  and  tell  how  much 
greater  the  average  is  than  19,  and  how  much  less  than  49. 

27.  The  averaore  of  27  and  55  is  how  much  more  than 
27  ?     How  much  less  than  55  ? 


REVIEW  207 

28.  To  find  the  average  of  3  numbers,  divide  their  sum 
by  3.     Find  the  average  of  20,  22,  and  24.     33,  36,  and  39. 

29.  To  find  the  average  of  4  numbers,  divide  their  sum 
by  4.     Find  the  average  of  21,  48,  72,  and  95. 

30.  Alfred's  %  in  an  arithmetic  test  was  95,  in  geogra- 
phy 94,  in  spelling  90,  and  in  writing  93.  What  was  his 
average  %? 

31.  What   was  the   average   age   of  the   children  of  a 

family  of  which  the  youngest  was  8  years  old,  the  next 

12,  the  next  15,  and  the  oldest  17? 

Show  that  ill  finding  the  average  of  numbers,  their  sum  is  divided 
by  the  number  of  them.  Make  class  exercises  by  averaging  the  ages 
of  different  groups,  or  their  standing  in  tests. 

32.  In  5  days  Fanny  worked  75  problems.  How  many 
did  she  average  a  day  ? 

33.  The  Blount  Plow  Works  made  12,345  plow  points 
in  June,  12,675  in  July,  and  12,945  in  August.  What  was 
the  average  made  in  the  summer  months  ? 

34.  If  twice  as  many  plow  points  were  made  in  Decem- 
ber as  in  June,  5555  more  in  January  than  in  July,  and 
3345  more  in  February  than  in  August,  how  many  plow 
points  were  made  in  the  winter  ?  What  was  the  average 
number  of  plow  points  made  in  the  winter  months  ? 

35.  Harold  earns  810.25  per  week,  Fred  earns  '111.50 
per  week,  and  Ernest  earns  $12.75  per  week.  How  much 
are  the  average  wages  of  the  boys  ? 

36.  14  companies  of  soldiers  have  1372  men  enrolled. 
What  is  the  average  number  in  a  company? 

37.  A  school  of  43  pupils  was  found  to  weigh  3483 
pounds.     What  was  the  average  weight  of  the  pupils? 

38.  The  combined  height  of  the  pupils  of  the  same 
school  was  172  feet.     What  was  the  average  height? 


20g  REVIEW 

39.  Use  71  as  a  divisor  and  as  dividends,  994,  6045,  3903. 

40.  With  2982  as  a  dividend  use  as  divisors,  71,  15,  28. 

41.  Wliat  is  the  quotient  when  884  is  the  dividend  and 
52  the  divisor  ? 

42.  What  is  the  quotient  when  1632  is  the  dividend 
and  51  the  divisor?  Multiply  the  quotient  by  51  and 
compare  it  with  1632. 

43.  What  is  the  quotient  when  2542  is  the  dividend 
and  62  the  divisor  ?  Multiply  the  quotient  by  the  divisor 
and  compare  it  with  the  dividend. 

44.  Find  quotient  when  3888  is  the  dividend  and  81 
the  divisor.  Compare  the  product  of  divisor  and  quotient 
with  dividend. 

Give  examples  similar  to  the  above  until  it  is  seen  that  when  there 
is  no  remainder  the  dividend  is  equal  to  the  product  of  the  divisor 
and  quotient.     Then  require  examples  proved. 

45.  2208 -f-  92  =  ?   1952-122  =  ?   3025 -f- 121  = 
8734  -  312  =  ?   4551  -^  123  =  ?   8988  -^  214  = 

46-       13  2     —  •     ~\Tl~  —  ■     ^12      —•         2  42  •       13  1 

47.  Find  quotient  when  697  is  the  dividend  and  21  is 
the  divisor.  Multiply  your  quotient  by  21,  add  the  re- 
mainder 4,  and  see  what  number  the  result  equals. 

By  oral  work  with  small  numbers  lead  the  class  to  see  the  method 
of  proof  in  this  case. 

Find  quotient  and  remainder,  and  prove  : 


Dividend. 

Divisor. 

Dividend. 

Divisor 

48. 

3839 

142 

53. 

4839 

156 

49. 

15699 

215 

54. 

17898 

213 

50. 

4294 

126 

55. 

5307 

221 

51. 

5782 

134 

56. 

5808 

215 

52. 

4879 

212 

57. 

15413 

214 

REVIEW 


209 


58.  When  2  oranges  can  be  bought  for  5  cents,  how 
much  will  1  orange  cost  ?     3?     7?    11?     12?     20?    40? 

59.  How  many  cubic  inches  in  a 
two-inch  cube  ?  Draw  a  picture  of  it. 
HoAV  many  square  inches  in  all  its 
surfaces  ? 


60.    How  many  lb.  in  a  ton  ?     In  2 J 
tons  ?     7|-  tons  ? 


Fig.  1 


61.    A   farmer  sold  5-J  tons  of   hay 
at  ^  10  a  ton.      How  much  did  he  receive  ?      He  bought  a 
wagon  for  $  48.25.     How  much  did  he  have  left  ? 


62.    Find  sums  : 

63.    Fi] 

ad  diffe 

rences 

• 

91        8J        Si 

^    n    If 

H        61         9^ 

5f 

381 
29^ 

or  1 

81* 

29i 

64.    From     8           2 

6 

5 

6 

7 

8 

take     6-J         0,- 

^ 

!i 

!i 

5i 

H 

65.    Multiply  :     3  J 

7 

4 

If 

7 

21i 

8 

11 

96^ 
12 

66.  1  ft.  is  what  part  of  a  yd.?  1  in.  is  what  part  of  a 
ft.?     ^2  of  J  of  a  yd.  is  what  part  of  a  yd.? 

67.  How  many  feet  does  a  man  walk  who  A\'alks  three 
times  around  a  lot  4  rods  square  ?     Draw  diagram. 

68.  A  night  watchman  has  the  duty  of  walking  four 
times  each  night  around  a  lot  that  is  18  rd.  long  and 
15  rd.  wide.     Hoav  many  ft.  does  he  Avalk? 

HORN.    ARITII.  14 


210  KEVIEVV 

69.  Draw  a  picture  of  a  3-inch  cube.  How  many 
cu.  in.  in  a  3-inch  cube  ?  How  many  sq.  in.  in  all  its 
faces  ? 

70.  What  is  the  ratio  of  6  to  3,  or  how  many  times  does 
6  contain  3  ? 

71.  Draw  a  line  3  inches  long  and  divide  it  into  inch 
lines.  Each  inch  line  has  what  ratio  to  the  3-inch  line  ? 
A  line  2  inches  long  has  what  ratio  to  the  3-inch  line  ?  A 
line  4  inches  long  has  what  ratio  to  the  3-inch  line? 
A  line  6  inches  long  has  what  ratio  to  the  3-inch  line? 
What  does  f  equal? 

By  illustration  with  lines  of  different  lengths  lead  children  to  see 
that  the  ratio  of  a  larger  number  to  a  smaller  one  is  the  quotient  of 
the  larger  divided  by  the  smaller.  Later  they  may  be  shown  that 
every  ratio  is  a  quotient. 

72.  What  is  the  ratio  of  12  to  3  ?     21  to  3  ?     25  to  3  ? 

73.  6  apples  will  cost  how  many  times  as  much  as  3 
apples  ?  If  3  apples  cost  2  cents,  how  much  will  6  apples 
cost?     9  apples?     15  apples? 

74.  When  corn  is  #.42  a  bushel,  how  many  bushels 
canbebought  for  17.56?     '$  13.02?     .^14.70?     #10.08? 

75.  How  many  cubic  inches  in  a  4-incli  cube  ?  Draw  a 
picture  of  one  and  tell  how  many  sq.  in.  in  its  surfaces. 

76.  How  many  2-inch  cubes  can  a  4-inch  cube  be  divided 
into  ? 

77.  What  is  the  ratio  of  20  to  4  ?     36  to  4  ?     48  to  4  ? 

78.  What  is  the  ratio  of  576  to  12?  1728  to  12? 
996  to  12  ? 

79.  8  apples  will  cost  how  many  times  as  much  as  4 
apples  sold  at  the  same  rate  ?  If  4  apples  cost  5  cents, 
how  much  will  8  apples  cost  ?     12  apples  ?     16  apples  ? 


REVIEW  211 

80.  5  is  one  of  a  pair  of  factors  that  make  55.  What 
is  the  other  factor  ?  Find  the  factor  that  helps  12  to  make 
516. 

81.  How  many  yd.  in  6  rd.  ?    8  rd.  ?    3  rd.  ?    11  rd.  ? 

82.  How  many  nickels  equal  35  cents  ?     i.75?     $.93? 

83.  Name  four  numbers  of  which  5  is  a  factor. 

84.  When  a  number  is  a  factor  of  two  or  more  numbers, 
it  is  called  a  Common  Divisor  of  them.  Name  a  common 
divisor  of  6  and  9,  12  and  8,  15  and  20,  25  and  35, 
21  and  35,  18  and  20,  18  and  27,  14  and  21,  15  and  20. 

85.  Turn  to  the  number  table  of  sevens  and  tell  what 
number  is  a  common  divisor  of  all  the  multiples  of  7  that 
are  in  the  table. 

86.  Name  two  numbers  that  have  a  common  divisor, 
and  tell  what  it  is. 

87.  Name  three  numbers  that  have  a  common  divisor, 
and  tell  what  it  is. 

88.  Turn  to  a  number  table  and  look  at  the  numbers 
whose  unit  figure  is  5.  What  number  is  a  common  divisor 
of  them  all  ? 

89.  Give  a  number  that  is  a  common  divisor  of  all  the 
numbers  whose  unit  figure  is  0. 

90.  What  number  is  a  common  divisor  of  all  the  even 
numbers  ? 

91.  Name  a  common  divisor  of  all  the  numbers  that  are 
printed  in  heavy  type  in  the  number  table  on  p.  114. 
On  p.  122.     On  p.  142.     On  p.  180. 

92.  Write  out  all  the  pairs  of  factors  that  make  20, 
and  all  those  that  make  45,  and  tell  which  is  the  greatest 
divisor  that  is  common  to  20  and  45. 


212  REVIEW 

93.  Find  three  common  divisors  of  12  and  18.  Which 
is  the  greatest  ? 

94.  Find  two  common  divisors  of  12  and  20,  and  tell 
which  is  the  greatest. 

95.  Make  a  list  of  common  divisors  of  20  and  40,  and 
tell  which  is  their  greatest  common  divisor. 

96.  Make  a  list  of  the  divisors,  and  pick  out  the  great- 
est common  divisor  of  20  and  30,  15  and  24,  30  and  40, 
24  and  30,  24  and  36,  25  and  30,  36  and  40,  35  and  49. 

97.  What  is  the  ratio  of  30  to  5  ?     40  to  5  ?     45  to  5  ? 

98.  10  hats  will  cost  how  many  times  as  much  as  5 
hats  ?  20  hats  Avill  cost  how  many  times  as  much  as  5 
hats  ?  If  5  hats  cost  $  3,  how  much  will  10  hats  cost  ? 
20  hats  ?     30  hats  ?     35  hats  ? 

99.  How  many  cubic  inches  in  a  5-inch  cube  ?  How 
many  square  inches  in  its  faces  ? 

100.    Add:  If    61   If       101.    From  61|   94f   27      16 
7^-   2|   8f  take  38J   37|   JJ   _2| 

102.  A  bolt  of  cloth  contained  37|  yd.     When  121  yd. 
were  sold,  how  many  yards  remained  ? 

103.  Multiply     751        85f        28f        27^        4|        6f 

by       7'  9  15  6  7  5 

104.  If  $1285.75  is  divided  among  5  men,  how  many 
dollars  and  cents  will  eacli  man  receive  ? 

105.  How  many  sixths  of  a  pie  in  2  pies  and  i  of  a  pie  ? 
In  31  pies  ?     51  ?     7f  ?     81  ?     lOf  ?     9i  ?     20i  ?     30f  ? 

106.  From  17      18      24  107.    Multiply  61   81   8f 

take     21     41   ^  ^^  L  L   L 

108.    Find  1  of  1248.66,     Of  1366.72.     Of  1968.22 
Of  11575.36. 


REVIEW  218 

109.  At  6%,  what  is  the  interest  of  a  dollar  for  6 
months,  or  i  a  year?  1  year  and  6  months?  2  years 
and  6  months  ? 

110.  How  many  seconds  in  i  a  minute  ?  In  ^  ?  In  J  ? 
How  many  minutes  in  one  quarter  of  an  hour?  In  |  of 
an  hour  ?     In  i  of  an  hour  ? 

111.  How  many  cu.  in.  in  a  6-inch  cube  ?  Into  how 
many  3-inch  cubes  can  a  6-inch  cube  be  divided  ? 

112.  How  many  square  inches  of  paj)er  would  it  take  to 
cover  a  box  in  the  shape  of  a  cube,  each  side  of  which  is 
6  inches  ? 

113.  Write  all  the  pairs  of  factors  that  make  36,  and 
those  that  make  48,  and  pick  out  the  greatest  number 
that  is  a  divisor  of  both. 

114.  In  the  same  way  find  the  greatest  common  divisor 
of  42  and  54,  36  and  60,  48  and  72. 

115.  What  is  the  ratio  of  18  to  6  ?     30  to  6  ?     6  to  54  ? 

116.  18  pencils  will  cost  how  many  times  as  much  as  6 
pencils  ?  If  6  pencils  cost  5  cents,  how  much  will  18 
pencils  cost  ?     30  pencils  ?     24  pencils  ?     42  pencils  ? 

117.  How  much  is  |  oi  ^?     i  of  ^  ?     |  of  i  ?     -^^  of  |  ? 

118.  HoAV  many  times  are  4  and  6  each  contained  in 
their  least  common  multiple  ? 

119.  How  many  days  in  3|  weeks  ?  In  4|  weeks  ?  In 
Sf  weeks  ?     In  9|-  weeks  ? 

120. 


122. 


Add:     8|.     5f 

8f      121.    From     82f     92       54 

fi2       76 

5f                 take     46f     57|    49f 

Multiply      51 
by     8 

9f         7f         8|        21j         96| 
6           7           8           22           12 

214  REVIEW 

123.  7  men  have  equal  shares  in  a  gokl  mine.  They 
take  from  it  in  one  year  f  17,355  worth  of  gold.  How 
much  is  each  man's  share  ? 

124.  At  7%  how  much  is  the  interest  of  a  dollar  for  5 
years  ?     2i  years  ?     8  years  and  6  months? 

125.  At  7%  interest  how  long  would  it  take  a  dollar  to 
gain  28  cents  ?     42  cents  ?     84  cents  ?     63  cents  ? 

126.  How  many  sq.  in.  in  a  rectangle  7  in.  long  and 
5^  in.  wide  ? 

127.  How  many  sq.  in.  in  a  right  triangle  whose  base 
is  7  in.  and  altitude  5^  in.? 

128.  How  many  cu.  in.  in  a  7-inch  cube  ?  How  many 
sq.  in.  in  all  its  surfaces? 

129.  iNIake  lists  of  all  the  factors  of  42  and  56,  and  find 
the  greatest  number  that  is  contained  in  both  of  them. 

130.  In  the  same  way  find  tlie  greatest  common  divisor 
of  42  and  63,  35  and  70,  21  and  42,  28  and  56. 

131.  Name  a  multiple  of  7  that  is  a  perfect  square. 
What  is  its  square  root  ? 

132.  How  much  is  1  of  4^  ?     f  of  |  ?     |  of  f  ?     f  of  f  ? 

133.  What  is  the  ratio  of  7  to  14  ?  7  to  42  ?  35  to  7  ? 
49  to  7  ?     28  to  7  ?     63  to  7  ? 

134.  14  tops  will  cost  how  many  times  as  much  as  7 
tops  of  the  same  kind  ?  If  7  tops  cost  10  cents,  how 
much  will  14  tops  cost  ?     21  tops  ?     35  tops  ?     28  tops  ? 

135.  Name  two  numbers  whose  greatest  common  divisor 
is  8.     Tell  how  many  times  8  is  contained  in  each  of  them. 

136.  Name  three  numbers  whose  greatest  common  divi- 
sor is  8,  and  tell  how  many  times  each  of  them  contains  8. 

137.  Name  a  multiple  of  8  which  is  a  perfect  square. 
What  is  its  square  root  ? 


REVIEW  215 

138.  How  many  quarts  in  3  pecks  ?  In  a  bushel  ?  In 
51  pk.?     lOfpk.?     41  pk.?     7|pk.?     201  pk.? 

139.  How  many  pecks  in  9  quarts?  21  qt.?  37  qt.? 
46  qt.?     58  qt.?     63  qt.?     77  qt.?     89  qt.? 

140.  How  many  8tlis  in  3  Avhole  ones  ?  In  41  ?  6J  ? 
7|?     5J?     9|?     12|?     8f? 

141.  Mr.  Kent  works  in  a  factory  where  8  hours  make 
a  day's  work.  How  many  hours  does  he  work  in  a  week  ? 
In  10  weeks  ? 

142.  How  many  sq.  in.  in  an  8-inch  square  ?  In  a  rec- 
tangle whose  base  is  8  in.  and  altitude  6|  in.?  In  a  right 
triangle  whose  base  is  8  in.  and  altitude  6 J  in.? 

143.  How  many  cu.  in.  in  an  8-inch  cube  ? 

144.  How  many  sq.  in.  in  all  the  surfaces  of  an  8-inch 
cube  ? 

145.  Add:  411-  75|-      146.    From     28|     76f     9 

^  51|  or  I     31|  take     19^     41f     6^ 

147.  Find  product:  31i     8f     21|     6 J       41       6^       9 J 

5       3         4       2       12       13       14 

148.  Find  I  of  1245.76.     Of  $334.32.      Of  1676.24. 

Of -1^889.60.     Of  1498.48. 

149.  If  i  7288  were  divided  among  8  men,  how  much 
would  each  man  receive  ? 

150.  At  8%,  how  much  is  the  interest  of  a  dollar  for  5 
years?  7  yr.  and  6  mo.?  9  yr.  and  6  mo.?  11  yr.  and 
6  mo. 

151.  How  much  is  1  of -1  ?     f  of  f  ?     foff?     fof|? 

152.  What  is  the  ratio  of  8  to  16  ?    48  ?    32  ?    96  ?    72  ? 

153.  What  is  the  ratio  to  8  of  48  ?    64  ?   40  ?    56?    88  ? 


216  REVIEW 

154.  24  pencils  will  cost  how  many  times  as  many  cents 
as  8  pencils  ?  When  8  pencils  are  sold  for  4  cents,  how 
many  cents  will  24  pencils  cost  ?    40  pencils  ?    5Q  pencils  ? 

155.  How  many  sq.  ft.  in  3  sq.  yd.?  6|-  sq.  yd.?  3 J 
sq.  yd.?     41  sq.  yd.?     7 J  sq.  yd.?     8f  sq.  yd.? 

156.  If  there  are  9  squares  in  a  row,  how  many  rows 
are  needed  to  make  a  rectangle  containing  54  squares  ? 
72  squares  ?     96  squares  ?     63  squares  ? 

157.  What  is  the  square  of  9  ?  What  is  the  square 
root  of  9  ?     Of  what  number  is  9  the  square  root  ? 

158.  How  many  cu.  in.  in  a  9-inch  cube  ? 

159.  How  many  sq.  in.  in  the  whole  surface  of  a  9-inch 
cube  ? 

161.    From  9^      9f      76^ 
take  1|      4f      29| 

162.  Florence's  mother  bought  33  yd.  of  calico,  and 
used  11^  yd.  in  making  a  dress.    How  many  yd.  were  left? 

163.  Make  problems  in  which  you  use  fractions.     ~ 

164.  Multiply:    91     9f     7f     If      9|      27^     38^     29i 

_4_5_6_7      ^       24       29         9 

165.  There  are  18^  acres  in  a  farmer's  tiela.  How  many 
acres  would  there  be  in  27  such  fields  ? 

166.  Find  quotients :     9)1758.79  9)1239.71 

9)1998.47  9)1621.34 

167.  9  boys  owned  a  boat  worth  1218.70.  How  much 
was  each  boy's  share  worth  ? 

168.  Write  two  numbers  of  which  9  is  the  greatest 
common  divisor,  and  tell  the  number  of  times  9  is  contained 
in  each. 


160. 

Add: 

n 

2^ 

8f 

n 

H 

8| 

n 

5| 

n 

REVIEW  217 

169.  A  man  bought  a  lot  for  #500,  built  a  house  for 
11000,  and  a  stable  for  1 200.  He  sold  the  property  for 
f  3560.     Did  he  gain  or  lose,  and  how  much  ? 

170.  27  marbles  will  cost  how  many  times  as  much  as  9 
marbles  at  the  same  rate  ?  When  9  marbles  are  sold  for 
5  cents,  how  much  will  27  marbles  cost  ?     36  ?     63  ?     81  ? 

171.  Name  the  multiples  of  9  until  you  reach  one  that 
is  also  a  multiple  of  6.  What  name  is  given  to  the 
smallest  number  that  exactly  contains  both  9  and  6  ? 

172.  Find  the  least  common  multiple  of  9  and  5,  9  and 
4,  9  and  8. 

173. 
174. 

'  10 


How  much  is 

1 

2 

01 

1? 
9  • 

^  of  ^  '^      ^  of  ^  '''     f  of  ^  '^ 

Find  sums : 

175. 

Find  differences : 

6A      »^ 
1t^      2^ 

Q  9_ 
^10 

9  7 
^10 

7i»o       6J^       9^       S^^ 
h\      n        51         4f 

9 

^10 

4_1_  9_9_  Q_9_ 

^10  ^10  ^10 

176.  Mr.  Wilson  had  24^9^  acres  of  land,  and  sold  21 1 
acres.     How  many  acres  had  he  left  ? 

177.  Multiply     35         375         25  15  13 

by     _10       _10       _100         1000         10000 

178.  Give  a  short  way  of  multiplying  a  number  by  10. 
By  100.     By  1000.     Make  some  examples  and  explain. 

179.  Multiply     24        41  82  51  212 

by     _20      _300      ^00        5000  30000 

180.  Make  some  examples  like  the  above,  and  tell  how 
you  multiply  when  the  multiplier  ends  in  naughts. 

181.  Find  products  : 

339^  462J^  596J^  463^3_   558_7_   661^2^   287-^0 
10    120     20     30     10     50     60 


218  REVIEW 

182.  If  it  takes  5^-q  yd.  of  bunting  to  drape  a  window, 
how  many  yd.  will  it  take  for  8  windows? 

183.  If  it  takes  1-^-q  yd.  of  bunting  to  drape  a  door,  how 
many  yd.  will  it  take  for  4  doors  ? 

184.  Divide:  10)3270. 

After  dividing  in  the  usual  way,  lead  the  children  to  see  that  the 
same  result  will  be  obtained  by  cutting  off  the  naught  in  the  units' 
place. 

185.  Divide  by  10  in  the  shortest  way :  4280,  3270, 47500. 

186.  Make  examples  and  show  how  you  divide  by  10 
any  number  that  ends  in  naught. 

187.  Make  examples  and  show  how  you  would  divide 
by  100  any  number  that  ends  in  2  naughts. 

188.  Make  examples  and  show  hoAv  you  can  divide  by 
1000  any  number  that  ends  in  3  naughts. 

189.  At  6  cents  a  square  foot,  how  much  will  it  cost  to 
sod  a  square  yard  of  the  lawn?     15  sq.  yd.  ?     30  sq.  yd.  ? 

190.  If  the  binding  used  costs  10  cents  a  foot,  how  much 
will  it  cost  to  bind  a  rug  1  yd.  square  ?     2  yd.  square  ? 

191.  If  you  put  $10  into  a  bank  that  pays  3%,  and  take 
none  out,  how  much  will  you  have  in  the  bank  at  the  end 
of  6  years?     12  years?     18  years? 

192.  Mary  bought  9  yd.  of  lace  at  8  cents  a  yd.,  and 
handed  the  clerk  75  cents.  How  much  change  should  she 
receive  ? 

193.  Make  problems  about  buying  and  making  change. 

194.  A  boy  picked  2  gal.  of  berries  on  Saturday  and 
3  gab  on  Monday.  He  sold  them  for  10  cents  a  qt.  How 
much  did  he  get  for  them  ? 

195.  Divide  12078  by  {  of  236. 


REVIEW  219 

196.  Name  three  numbers  of  which  10  is  the  greatest 
common  divisor. 

197.  Find  factors  of  40  and  60,  and  pick  out  the 
greatest  common  factor. 

198.  Name  in  order  the  multiples  of  9  until  you  find 
one  that  is  also  a  multiple  of  4.  By  it  divide  432,  1296, 
and  2592. 

199.  What  is  the  least  common  multiple  of  10  and  4  ? 
10  and  7  ?     10  and  6  ?     10  and  8  ?     10  and  12  ? 

200.  2  ^-^  iV  ~  ^  2  ^^  "iV  of  a  dollar  =  how  many  cents  ? 
^  of  ^  of  a  dollar  =  how  many  cents  ?  ^  of  -^^  of  a 
dollar  =  how  many  cents  ? 

201.  40  oranges  will  cost  how  many  times  as  much  as 
10  oranges  ?  If  10  oranges  cost  25  cents,  how  much  will 
40  oranges  cost  ?     20  ?     50  ?     TO  ?     90  ? 

202.  How  many  cu.  in.  in  a  10-inch  cube  ?  How  many 
sq.  in.  in  its  surface  ?  How  many  in.  long  are  all  its 
edges  taken  together  ? 

203.  How  many  sq.  in.  in  a  rectangle  10  in.  long  and 
9^  in.  wide  ?     How  long  is  its  perimeter  ? 

204.  The  perimeter  of  a  square  is  40  in.  How  long  is 
one  side  ?     How  many  sq.  in.  in  the  square  ? 

205.  A  rectangle  is  10  in.  long  and  its  perimeter  is 
.30  in.  How  wide  is  the  rectangle  and  what  is  its  area? 
Draw  diagram. 

206.  John  may  draw  a  rectangle,  not  letting  any  one 
else  see  it.  He  may  give  its  length  and  the  length  of  its 
perimeter  to  the  class.  They  may  find  the  width  of  the 
rectangle  and  its  area. 


220  REVIEW 

207.  Mary  may  draw  a   rectangle   and  give  its  width 

and  the  length  of  its  perimeter.     The  class  may  find  the 

length  of  the  rectangle  and  its  area. 

Let  this  exercise  be  general.  Encourage  the  children  to  dispense 
with  the  drawing  of  the  figure  as  soon  as  they  are  able  to  visualize  it 
clearly. 

208.  What  do  you  mean  by  -^  of  anything  ?    Illustrate. 

209.  -^j  of  a  yd.  of  cloth  and  -^j  of  a  yd.  and  ^^  of  a 
yd.=  how  many  whole  yd.? 

210.  How  many  llths  in  2  whole  ones  ?     In  3  ?     4^  ? 

211.  How  many  whole  ones  in  |^  ?    if  ?    ||  ?    ^  ?    ^  ? 

212.  Find  sums  :  213.    Find  differences  : 

3A        h\        hS         ■t"T\        7  6  6 

5A        lA        8j^  8A        3tV        h\        lA 

214.  From  a  piece  of  goods  containing  S^j  yd.,  o^j 
yd.  were  cut  off.     How  much  remained? 

215.  Write  products  : 

11    11     11    11    11    11    11    11    11 

81    Q  1     91    ^1    ^1    ^1    71    .'^i    4  2_ 

__2    ^'tt   _%   _%   _^  _2a      __1   __I   ^tt 

216.  How  many  sq.  in.  in  a  rectangle  11  in.  long  and 
7^  in.  wide?     How  long  is  its  perimeter? 

217.  Find  area  and  perimeter  of  a  rectangle  11  in.  long 
and  3J  in.  wide. 

218.  How  many  sq.  in.  in  a  right  triangle  whose  base 
is  11  in.  and  altitude  8  in.? 

219.  How  many  cu.  in.  in  an  11-inch  cube  ?  How  many 
sq.  in.  in  all  its  surfaces  ?  AVhat  is  the  length  of  all  its 
edges  taken  together? 

220.  Write  three  numbers,  of  which  11  is  the  greatest 
common  divisor. 


REVIEW  221 

221.  How  many  sq.  in.  in  a  square  foot  ?  How  many 
sq.  ft.  in  1584  sq.  in.?     3024  sq.  in.?     4752  sq.  in.? 

222.  How  many  cu.  in.  in  a  cubic  foot?  How  many 
cu.  ft.  in  19,008  cu.  in.?     20,736  cu.  in.?     25,920  cu.  in.? 

223.  What  is  the  least  multiple  of  11  that  will  contain 
3?     7?     5?     8? 

224.  How  much    is  J  of  3^^  ?     |  of  -f^  ?     |  of  y\  ? 

225.  Use  11  as  a  divisor  with  462,  484,  572,  683,  782. 

226.  The  expense  of  an  excursion  which  cost  S  374  was 
shared  equally  by  11  men.     How  much  did  each  man  pay  ? 

227.  33  yd.  of  cloth  will  cost  how  many  times  as  much 
as  11  yd.  at  the  same  rate?  If  11  chocolate  drops  cost  5 
cents,  how  many  can  be  bought  for  10^?     For  15^? 

228.  How  much  is  f  of  12  ?     |  of  12  ?     |  of  12  ? 

229.  Multiply  12  by  OJ.     By  ^,     5f     2i.     3^. 

230.  How  many  12ths  in  3  whole  ones  ?    In  4^2  -     ^ii  ^ 

f,_5^'?       ^J^9       6_7_9       Q1JL9 

231.  How  many  whole  ones  in  J^J  ?     In -H  ?    U?    ^|^? 

232.  Find  sums  :  233.    Find  differences  : 

8tV  6^       211  811       61  or  3^     9^       7i 

61  or  ^2       ^12       Hi  5  7  9  8 

9  1_  71  41*'  IJL         B-5-  4  J-        411 

""12  *^  ^1  ^12  ^12  ^12  ^12 

234.  A  farmer  had  96\^  acres  of  woodland  and  238^2 
acres  of  cleared  land.     How  many  acres  had  he  in  all  ? 

235.  To  3  ft.  and  4  in.  add  2  ft.  and  5  in.,  placing  the 

work  as  below. 

ft.  in. 

3  4 

2 5 

5  9  Ans. 


9 

•J 

REVIEW 

ft. 

in. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

236. 

Add: 

7 

4 

4 

11 

6 

8 

9 

8 

3 

7 

3 

1 

2 

5 

1 

7 

ft. 

in. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

21 

3 

16 

8 

8 

9 

10 

9 

8 

10 

5 

9 

2 

7 

4 

6 

ft. 

in. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

237. 

From 

11 

8 

9 

7 

8 

9 

12 

11 

take 

6 

2 

.    3 

4 

4 

^ 

D 

2 

7 

238.  If  you  have  a  string  4  ft.  long  and  cut  off  1  inch, 
how  many  ft.  and  in.  long  is  it  then?  How  long  is  it 
when  you  have  cut  off  2  ft.  more  ? 

fto        in.  ft.  in.  ft.        in.  ft.      in, 

239.  From     40  7         0  80  90 

take     21  23  46  38 


240.  A  room  is  12  ft.  high.  The  border  around  the 
tojj  of  the  wall  is  1  ft.  6  in.  wide.  How  far  is  the  lower 
edge  of  the  border  from  the  floor  ? 

241.  ft.    in. 

From     3      1  ^^^  ^^^^^  ^^  illustrated  by  measurement  if  nee- 

take    0    5    ^'"''^- 


242. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

ft. 

in. 

From 

6 

2 

9 

3 

8 

1 

6 

4 

9 

6 

take 

2 

4 

3 

5 

3 

7 

2 

7 

3 

8 

243.  Arthur  is  4  ft.  7  in.  tall  and  Mary  is  5  ft.  1  in. 
What  is  the  difference  in  their  heights  ? 

244.  How  tall  were  you  when  you  were  1  ft.  and  6  in. 
shorter  than  you  are  now  ? 

Make  a  general  exercise  by  having  the  children  measure  the 
lieights  of  their  classmates  or  of  different  objects  in  the  schoolroom, 
^nd  find  differences. 


ft. 

in„ 

12 

10 

3 

ft. 

in. 

9 

7 

4 

ft. 

in. 

10 

6 

6 

ft. 

in. 

5 

7 

5 

ft. 

in. 

9 

2 

8 

REVIEW  223 

245.  ft.    in.  ft.    in.  ft.    in,  ft.    in.  ft.    in. 

Multiply:  32  76  53  25  56 

5  2  4  3  4 

ft.     in. 

7  8 
2 

246.  How  long  is  the  perimeter  of  a  square,  one  side  of 
wliicli  is  5  ft.  and  4  in.?  4  ft.  7  in.?   6  ft.  3  in.?  2  ft.  11  in.? 

247.  The  long  sides  of  a  rectangle  are  each  4  ft.  and 

8  in.  long.     The  short  sides  are  3  ft.  and  4  in.     How  long 
is  the  perimeter  ? 

248.  Make  problems  about  the  perimeters  of  figures. 

249.  Mr.  Wilson  has  a  flower  bed  in  the  shape  of  a  six- 
pointed  star.  See  Fig.  3,  p.  201.  Each  side  of  the  points 
is  2  ft.  and  6  in.  long.  How  long  is  the  perimeter  of  the 
flower  bed  ? 

250.  How  long  is  the  perimeter  of  an  equilateral  tri- 
angle, one  side  of  which  is  5  ft.  and  1  in.  long.?    4  ft.  4  in.? 

251.  How  many  sides  has  a  hexagon  ?  On  what  l^age 
in  your  book  can  you  find  one  ? 

252.  Julia  has  a  flower  bed  in  the  shape  of  an  equi- 
lateral hexagon,  bordered  with  pinks  j)laced  1  ft.  apart. 
Each  side  of  the  border  is  2  ft.  long.  Draw  a  diagram  of 
the  flower  bed,  and  find  how  many  plants  are  in  the  whole 
border. 

253.  Add : 

yr.   mo.         yr.    mo.  yr.    mo.         yr.  mo.  yr    mo  yr.   mo. 


8  3 

9  6 

11  3 

4  11 

21  8 

31  7 

7  5 

8  8 

4  10 

7  3 

17  9 

12  11 

254.    Helen  is  11  yr.  and  7  mo.  old  and  Emma  is  4  yr. 
and  8  mo.  older  than  Helen.      How  old  is  Emma  ? 


224  REVIEW 

255.  3  yr.  and  7  mo.  ago  Edwin  was  8  yr.  and  9  mo. 
old.     How  old  is  he  now  ? 

256.  How  old  will  you  be  in  2  yr.  and  3  mo.  from  noAV  ? 

257.  Make  problems  about  ages  in  years  and  months. 


^oo. 

yr- 

mo. 

yr. 

mo. 

yr- 

mo. 

yr. 

mo. 

yr. 

mo 

From 

13 

6 

15 

9 

16 

11 

18 

0 

19 

0 

take 

7 

3 

12 

4 

5 

6 

5 

1 

4 

3 

259.  Albert  is  13  yr.  and  8  mo.  old.  How  long  before 
he  will  be  15  years  old? 

260.  yr,    mo.  yr.    mo.  yr.     mo.  yr.    mo.  yr      mo. 

From     36     1         48     2         37     7  24     5        31     4 

take       2     3         12     5         13     9  19     8        24     9 


261.  How  old  were  you  3  yr.  and  2  mo.  ago  ? 

262.    is  yr.   and  mo.  old,  and  is 

—   yr.    and  mo.    old.      Find    difference    between 


their  ages.  • 

For  a  general  class  exercise  compare  ages  of  different  members  of 
the  class,  disregarding  days. 

263.  yr.  mo.         yr.  mo.         yr.  rao.  yr.    mo.  yr.     mo. 

Multiply:   53         74         86         12     8         11     7 

3  3  4  3  3 


264.  Harriet  is  11  years  and  6  months  old.      Her  mother 
is  three  times  as  old.     How  old  is  her  mother  ? 

265.  Make  problems   in  which  years   and  months  are 
multiplied. 

266.  Divide  by  13  each  of  the  numbers  between  2000 
and  3000  that  end  in  97. 

267.  Mr.  Anderson  owns  -^^  of  a  mine  that  paid  one 
year  187,872.     How  much  did  he  receive? 

268.  The    next   year    the    mine    yielded    only    16472. 
How  much  did  he  receive  ? 


REVIEW  225 

269.  The  next  year  the  mine  lacked  $765  of  paying 
expenses.     How  much  did  Mr.  Anderson  have  to  pay  out  ? 

270.  Mr.  Brown  has  a  salary  of  $3500  a  year.  How 
much  does  he  receive  each  month  ? 

271.  If  he  saves  $125  every  month,  how  much  does  he 
save  in  a  year  ?     How  much  will  he  save  in  12  years  ? 

272.  A  dealer  in  wagons  paid  $564  for  a  dozen  wagons 
of  the  same  kind.     How  much  did  each  wagon  cost  ? 

273.  If  he  gained  $5.50  on  each  Avagon,  how  much  did 
he  gain  on  the  dozen  ? 

274.  Mr.   West,  who   has    a   stationery   store,    bought 

11  dozen  tablets  for  $3.96.  What  was  the  price  per 
dozen  ?  How  much  did  each  tablet  cost  ?  If  he  sells 
them  at  $.05  each,  how  much  does  he  gain  on  them  all? 
The  cost  of  one  is  in  what  ratio  to  the  gain  on  one  ? 

275.  What  is  the  ratio  of  21  to  12  ?     48  to  12  ?     72  to 

12  ?     36  to  12  ?     108  to  12  ? 

276.  24  bicycles  will  cost  how  many  times  as  much  as 
12  bicycles  of  the  same  kind  ?  If  a  dozen  bicycles  cost 
$600,  how  much  will  24  bicycles  cost  ?  36  ?  48  ?  96  ?  72  ? 

277.  If  a  dozen  bicycles  cost  $500,  how  many  can  be 
bought  for  $1500  ?  For  $2500  ?  For  $3500  ?  For  $4500  ? 
For  $2000? 

278.  Square  14,  98,  195,  117. 

279.  If  84  men  form  a  military  company,  how  many 
companies  can  be  formed  by  1092  men  ?     1764  men  ? 

Divide: 

280.  46968  by  206      283.  939695  by  815 

281.  88392  by  509      284.   12750  by  315 

282.  634876  by  411      285.   12750  by  316 

HORN.  ARITH.  15 


Dividends 

Divisors 

Dividends 

Divisors 

22260 

212 

292. 

84941 

841 

122811 

611 

293. 

178488 

888 

48569 

423 

294. 

48144 

472 

136841 

671 

295. 

71173 

691 

12750 

125 

296. 

198198 

18 

226  EEVIEW 

286.  Multiply  212  611  423  671  1228  1728  6843 
by  105  201  103  204  1004  1005  2001 

0  in  the  quotient  is  the  special  difficulty  of  the  following  examples. 
Find  quotients : 

287. 
288. 
289. 
290. 
291. 

297.  Mr.  Hunt  had  851^  acres  of  corn,  108J  acres  of 
wheat,  2J  acres  of  cabbage,  54  acres  of  oats,  13  acres  of 
potatoes,  IJ  acres  of  radishes,  and  15|-  acres  of  rye.  How 
many  acres  of  grain  had  he  ?  How  many  acres  of  vege- 
tables had  he  ? 

298.  What  is  the  ratio  of  385  to  35  ?     Of  462  to  42  ? 

299.  How  many  inch- cubes  would  it  take  to  cover  a 
square  foot  ?  How  many  layers  of  inch  cubes  to  build  a 
cubic  foot  ?     How  many  inch  cubes  to  build  a  cubic  foot  ? 

300.  How  many  cu.  ft.  in  a  coal  bin  which  is  10  ft. 
long,  8  ft.  wide,  and  7  ft.  high  ? 

301.  How  many  cu.  ft.  in  a  cellar  24  ft.  long,  20  ft. 
wide,  and  7  ft.  deep  ? 

302.  How  many  cu.  ft.  in  a  ditch  40  ft.  long,  5  ft. 
wide,  and  3  ft.  deep  ? 

303.  How  many  cu.  ft.  of  air  in  a  room  30  ft.  long, 
20  ft.  wide,  and  10  ft.  high  ? 

304.  Class   Exercise. may  think  of  a  room 

that  has  four  smooth  walls,  give  its  probable  length, 
breadth,  and  height.  The  class  may  find  how  many  cubic 
feet  it  contains. 


REVIEW  227 

305.  1  cent  is  what  part  of  1  dollar  ? 

306.  "Y-QQ  of  anything  is  sometimes  called  1%  of  it. 
What  shall  we  call  -^f  q  ?     -^ ?    y^o  •     i oo  ^^*  ^^^^  whole ? 

307.  Have  you  ever  stood  100%  on  an  examination  or 
test?     What  does  100%  mean? 

308.  If  you  lacked  2%  of  being  perfect  on  an  examina- 
tion, Avhat  %  would  you  stand  ? 

309.  When  a  man  loses  100  %  of  his  money,  wliat  %  of 
it  has  he  left  ? 

310.  7  cents  is  Avliat  %  of  a  dollar?  What  %  of  a  dol- 
lar is  9  cents?     13  cents  ?     21  cents?     99  cents? 

311.  $  17  is  what  %  of  #  100  ?    What  %  of  1 100  is  1 19  ? 

312.  9  inches  is  what  %  of  100  inches?  AYhat  %  of 
loo  inches  is  31  inches?     41  inches?     1  yard  and  1  incli? 

313.  If  you  get  6  cents'  interest  for  every  100  cents  you 
lend  for  a  year,  what  %.are  you  getting? 

314.  Mr.  Ta3lor  gets  5  cents'  interest  each  year  for 
every  dollar  he  lends.  What  %  does  he  get  ?  How 
much  interest  does  he  get  each  year  for  ^$7.00  ? 

315.  Point  out  numbers  on  the  number  table  and  tell 
what  %  they  are  of  100,  and  what  %  they  lack  of  being 
equal  to  100. 

316.  50  cents  is  what  part  of  a  dollar  ?  What  %  of  a 
dollar  ? 

317.  25  cents  is  what  part  of  a  dollar  ?  What  %  of  a 
dollar  ? 

318.  75  cents  is  what  part  of  a  dollar  ?  Wliat  %  of  a 
dollar  ? 

319.  I  of  100%  =  how  many  %  ? 

320.  How  much  is  50%  or  i  of  18  ?    50%  of  24  ? 


228  REVIEW 

321.  Turn  to  Fig.  2,  page  201,  and  show  50%  of  the 
figure.  Show  50%  of  Fig.  3,  page  201.  Of  Fig.  2, 
page  189. 

322.  George  had  f  10,  and  lost  50%  of  his  money. 
How  much  did  he  lose,  and  how  much  had  he  left  ? 

323.  Mr.  Hall  is  6  feet  high.  The  height  of  his  son 
Charles  is  50%  of  Mr.  Hall's  height.     How  tall  is  Charles? 

324.  Caroline's  age  is  50%  of  that  of  her  teacher,  who 
is  25  years  old.     How  old  is  Caroline  ? 

325.  How  much  is  50%  of  28?  280?  140?  360?  840? 

326.  50%  of  a  gallon  =  how  many  quarts?  50%  of  a 
pound  =  how  many  ounces  ?  50%  of  a  peck  =  how  many 
quarts?  50%  of  a  ton  =  how  many  pounds  ?  50%  of  a 
foot  =  how  many  inches?  50%  of  a  square  foot  =  how 
many  square  inches?  50%  of  a  cubic  foot  =  how  many 
cubic  inches?  50%  of  a  yard  =  how  many  feet?  How 
many  inches  ? 

327.  50%  of  a  square  yard  =  how  many  square  feet? 
How  many  square  inches  ? 

328.  3  is  50%  of  what?  7  is  50%  of  what?  11  is 
50%,  of  what?     13  is  50%  of  what? 

329.  John  had  8  cents,  and  gained  as  much  more.  How 
much  had  he  then  ?  How  much  would  he  have  had  if  he 
had  gained  only  50%  as  much  more  ? 

330.  Thomas  had  12  cents  and  gained  50%  more.  How 
much  had  he  then  ? 

331.    may  give  a  problem  to  the  class  about  some 

one  who  had  some  money  and  gained  50%. 

332.  Find  50%  of  10136  and  divide  it  by  24. 


REVIEW 


229 


Fig.  2 


333.  Find  50%  of  8148,  and  divide  it  by  32. 

334.  ^  of  100  %  of  anything  =  how  many  %  ? 

335.  Draw  a  circle  and  divide  it  into 
fourths.  Write  in  each  fourth  the  %  which 
it  is  of  the  whole  circle. 

336.  Draw  a  square  2  inches  long  and  show 
25%  of  it.  Show  25%  of  Fig.  5,  page  158. 
Of  Fig.  1,  page  157. 

337.  Henrietta  is  8  years  old,  and  her  little  sister's  age 
is  25%  of  hers.     How  old  is  her  little  sister? 

338.  Mr.   Adams  had  112  and  lost  25%  of  it.     How 
much  did  he  lose,  and  how  much  had  lie  left  ? 

339.  Find  25%  of  40.     24.     36.     248.     432.     888. 

340.  6  is  25%  of  what  number  ? 

341.  25  per  cent  of  what  number  is  3  ?     5  ?     7  ?     12  ? 

342.  Richard  had  20  cents  and  gained  25%.    How  much 
had  he  then  ? 

343.  Mr.  Walker  had  1400  and  lost  25%  of  it.     How 
much  had  he  then  ? 

344.  Make  problems  about  some  one  who  gamed  or  lost 
25%  of  a  sum  of  mone3^ 

345.  Find  25  %  of  34272  and  divide  it  by  43. 

346.  Find  25%  of  8028  and  divide  it  by  61. 

347.  When  a  man  loses  25  %  of  his  money,  what  %  has 
he  left  ?     How  many  fourths  of  his  money  are  left  ? 

348.  Draw  a  line  12  inches  long  and  show  75%  of  it. 

349.  Show  75%  of  Fig.  2,  page  189.     Of  Fig.  5,  page 
202. 

350.  Find  some  figures  in  the  book  that  you  can  show 

25%  of.     50%.     75%. 


230  REVIEW 

351.  75%  of  a  gallon  =  how  many  quarts?  75  %  of  a 
bushel  =  how  many  pecks  ?  75  %  of  a  pound  =  how  many 
ounces  ? 

352.  Draw  a  rectangle  8  inches  long  and  2  inches  wide, 
and  show  75  %  of  it. 

353.  Show  75  %  of  these  figures  : 

A  rectangle  8  in.  by  4  in. 
A  rectangle  4  in.  by  3  in. 
A  4-inch  square. 
A  rectangle  10  in.  by  4  in. 

354.  Ida's  age  is  75%  of  the  age  of  Ella,  who  is  12  yr. 
old.     How  old  is  Ida  ? 

355.  Mr.  Edwards'  horse,  Claybank,  sold  for  75  %  of 
the  price  of  another  horse  of  his  called  Redtop.  Redtop's 
price  was  $  400.     What  was  Claybank's  price  ? 

356.  William  had  20  cents  and  lost  75%  of  it.  How 
much  did  he  lose  ?     How  much  had  he  left  ? 

357.  Arthur  had  8  cents  and  gained  75%.  How  much 
had  he  then  ? 

358.  Thomas  had  20  cents  and  gained  75%.  How  much 
had  he  then  ? 

359.  Make  problems  in  which  75%  of  a  sum  of  money 
is  lost  or  gained. 

360.  Find  amount  of  the  following  bill : 

Boston,  Mass.,  Jan.  3,  1898. 
Mr.  Thomas  Reed, 

Bought  of  James  Wilson  and  Co., 

12  1b.  Coffee  at  $.30  ? 

4  lb.  Butter  at     .25  ? 

25  lb.  Sugar  at      .05  ? 

3  lb.  Starch  at      .15  ? 


REVIEW  231 

Get  billheads  from  local  merchants.  Let  children  make  imaginary 
purchases  of  one  another,  copying  the  billheads,  making  out  bills,  and 
receipting  them. 

361.  Lucy  Wood  is  going  to  the  country.  Her  mother 
bought  8  yards  of  gingham  at  ^.11  a  yard  to  make  her 
a  dress,  and  2  yards  of  lace  for  it  at  $.24  a  yard.  She 
paid  $1.15  for  the  makmg  of  the  dress.  How  much  did 
the  dress  cost  ? 

362.  She  bought  a  hat  for  $.50,  some  flowers  for  $.35, 
and  3  yd.  of  ribbon  at  $.18  a  yd.  She  paid  the  milliner 
$.50  for  trimming  the  hat.     How  much  did  the  hat  cost? 

363.  She  has  bought  a  rain  cloak  for  $3.75,  an  umbrella 
for  $  1.25,  and  a  pair  of  rubbers  for  $  .35.  How  much  has 
Mrs.  Wood  spent  to  keep  Lucy  from  getting  Avet  ? 

364.  hicluding  $2.50  for  a  pair  of  shoes  and  $2.75  for 
a  tennis  racket,  ho\v  much  has  Lucy's  whole  outfit  cost  ? 

For  a  class  exercise  let  pupils  find  cost  of  preparing  an  outfit  to 
go  camping,  to  go  to  the  city,  to  the  seashore,  to  a  picnic,  etc.  Let 
pupils  suggest  items  and  estimate  cost. 

365.  Divide  2025  by  the  square  root  of  144. 

366.  Divide  89286  by  the  9th  multiple  of  8. 

367.  Divide  30292  by  the  8th  multiple  of  9. 

368.  Divide  1487  by  the  least  common  multiple  of  9 
and  <S. 

369.  George  Washington  was  born  in  MDCCXXXII. 
How  old  was  he  in  the  year  in  which  the  Declaration  of 
Independence  was  signed  ? 

370.  Mary  has  a  flower  bed  in  the  form  of  a  square  with 
6  rose  bushes  on  each  side  of  the  bed,  but  there  are  not  24 
rose  bushes  in  it.  How  many  are  there  ?  Make  dots  to 
shoAv  the  position  of  the  rose  bushes  and  count  them 
unless  you  can  tliink  how  tliey  look. 


CHAPTER  XVII 


FRACTIONS 

1.    Copy  Fig.  1  by  drawing,  making  each  side  of  the 

figure  4  inches  long.  How  man}^ 
inch  squares  in  your  figure  ?  Each 
square  is  what  fractional  part  of  the 
figure  ? 

2.  3  squares  are  what  part  of  the 
figure  ?  5  squares  ?  7  squares  ? 
9  squares?     13  squares? 

3.  Show    i    of    Fig.     1.      How 


Fig.  1 


many  16ths  in  ^? 


4.    Fill  out  the  following  by  studying  the  parts  of  Fig.  1 : 


1  _   1 

1  _     ? 

3  _ 

? 

1    _     ? 

3  _     ? 

5  _     ? 

2  —  TB"- 

4  —  "TS"- 

¥  — 

1  6- 

8  ~~   1 6- 

8  -"   16- 

8  —  IF- 

5.  Show  i  of  Fig.  1.     i  of  Fig.  1.     i  of  Fig.  1. 

6.  Show  ^  of  ^  of  the  figure.  What  part  of  the  figure 
is  ^  of  ^  of  it  ? 

7.  Show  ^  of  ^  of  Fig.  1.  What  part  of  the  figure  is 
J  of  |;  of  it  ?  Show  -\  of  J  of  the  figure.  What  part  of 
the  whole  is  it  ? 

8.  Show  ^  of  J  of  Fig.  1.      How  much  is  -j  of  |^? 

9.  Show  ^  of  P'ig.  1.     Sliow  ^  of  ^  of  it.     i  of  ^  =  ? 

10.    Show  J  of  1^  of  it.      What  part  of  Fig.  1  is  ^  of  | 
of  it  ? 

232 


FRACTIONS 


233 


Fig.  3 


11.  Change  your  copy  of  Fig.  1  into  a  copy  of  Fig.  2 
by  drawing  oblique  lines. 

12.  How  many  right  triangles 
in  Fig.  2  ? 

13.  Each  square  is  what  part  of 
Fig.  2  ?  Each  triangle  is  what 
part  of  a  square  ?  Each  triangle 
is   what   part   of   Fig.    2?     J   of 

JL  — 9 
16  ~- 

14.  The  figure  ABCD  is  what 
part  of  the  whole  figure  ?  One 
of  the  triangles   is  what  part  of 

ABCD?     A  triangle  is  what  part  of  the  whole  figure? 
1  of  J-  =  '? 

8   ^^   4       • 

15.  What  name  is  given  to  a  fraction  of  a  fraction? 
How  do  you  find  the  value  of  a  fraction  of  a  fraction  ? 

Let  children  apply  the  rule  to  the  preceding  questions,  and  show 
that  the  same  results  are  obtained  by  following  the  rule  as  by  actually 
dividing  the  figures  and  counting  their  parts. 

16.  A  single  fraction  is  called  a  Simple  Fraction.  Ex- 
press J  of  -^Q  by  a  simple  fraction. 

17.  Change  the  following  to  equivalent  simple  fractions  : 
1  of  1    1  of  5.    2  of  6    A  of  ^ 

Explain  the  term  '^equivalent  fractions." 


18. 


Which  is  greater  |  of  f  or  J-^  ?     f  of  #  or  Jf  ? 


of  S.  ov  2.7.  ? 
'^'^  7  ^     35  • 

19.  Ernest  rode  J  mile  on  his  bicycle,  and  Gertrude  rode 
J  as  far  as  he  rode.  What  part  of  a  mile  did  Gertrude 
ride  ? 

20.  Make  problems  that  give  compound  fractions  and 
find  their  value  in  simple  fractions. 


234  FRACTIONS 

21.  Reduce  the  following  to  equivalent  simple  fractions  : 

3  of  2_  4  of  -I  8  of  1-4  2  of  18.  6  of  11  5  of  lA  i.  of  2JL  5 
y  Ol  g,  ,^  Ui  g,  Y  Oi  24^  9  *J^  20'  7  '^^  18'  7  ^^  15'  9  ^^  72'  6 

of  1  2  3  of  2  8  5  of  3-  of  12  2.  of  3.  of  i-  of  4 
^^  15'  Y  ^^  2^'  6  ^^  5  ^^  15'  3  *^^  4  ^^  5  ^^  T* 

22.  1  off  of  27  =  ?      I  off  of  28  =  ?      |of^\of22  =  ? 

23.  Class  E:^EticiSE. may  give  a  compound  frac- 
tion, and  the  class  may  reduce  it  to  an  equivalent  simple 
fraction. 

24.  How  much  is  1  of  ^  of  a  square  yard  ?  How  many 
square  feet  ? 

25.  What  part  of  a  square  foot  is  ^2  of  3^2  ^^  ^^  ^  How 
many  square  inches  ? 

26.  John  may  start  from  one  side  of  the  schoolroom  and 
walk  a  distance  which  he  thinks  is  a  rod.  Another  pupil 
may  measure  the  distance  to  see  how  nearly  right  he  is. 

27.  320  rods  make  a  mile.  How  many  rods  in  1  of  a 
mile?     I?     f?     I? 

28.  How  many  rods  in  f  of  -f^  of  a  mile  ?  In  J  of  |  of 
y^g  of  a  mile  ? 

29.  Mrs.  Adams  has  money  enough  at  interest  to  give 
her  i288  interest  every  year.  How  much  does  her  money 
earn  in  6  mo.  ?     3  mo.  ?     9  mo.  ? 

30.  If  she  had  only  |-  as  mucli  money  at  interest,  how 
much  would  she  get  from  it  each  year  ? 

31.  Her  money  is  now  at  6%  interest.  If  she  had  just 
as  much  money  at  interest,  hut  the  rate  was  only  1  %,  how 
much  interest  would  she  receive  each  year  ?  How  mucli 
is  the  interest  of  the  same  money  for  the  same  time  at  3  %  ? 
5%?     7%?     9%? 

32.  Mr.  Smith  has  just  collected  |872  interest  from  a 
note  that  had  been  gaining  interest  for  8  years.  How 
much  did  it  gain  each  year  ?  If  it  had  been  paid  a  year 
ago,  how  mucli  interest  should  he  have  received? 


FRACTIONS  235 

33.  Make  problems. 

34.  Recopy  Fig.  2.     Each  square  is  what  part  of  Fig.  2  ? 
Each  triangle  is  what  part  of  Fig.  2  ? 

35.  ^  of  Fig.  2  equals  how  many  16ths  of  it  ?     How 
many  32ds  of  it  ? 

36.  Show  ^  of  Fig.  2.     How  many  16ths  in  |  of  Fig.  2  ? 
How  many  32cls  in  ^  of  the  figure  ? 

37.  Show  ^  of  Fig.   2,  and  tell  how  many  16ths  of  it 
there  are  in  |^.     How  many  32ds  of  it  there  are  in  ^  ? 

38.  Fill  out  the  following  by  studying  the  parts  of  Fig  2 : 


1  _  ? 

2         32' 

9 

16 

9 

i  —    ? 

8  ~"  32' 

? 
16  • 

¥  ~  32' 

•> 

T6^ 

9 
t' 

3  _     ? 
8  —  32' 

9 

T6- 

3  ? 

4  —  32' 

9 

9 

5  _  JL 
8         32' 

? 
16* 

K  the  children  have  not  discovered  it  for  themselves,  show  them 
that  the  method  of  reducing  a  fraction  to  lower  terms  by  dividing  both 
terms  of  it  by  the  same  number,  or  raising  it  to  higher  terms  by  mul- 
tiplying both  terms  by  the  same  number,  gives  the  same  result  as  by 
dividing  figures  and  counting  the  parts. 

39.  ^  of  anything  equals  how  many  6ths  of  it  ?  How 
many  8ths  of  it?  How  many  lOths  ?  How  many  12ths  ? 
How  many  14ths  ?  How  many  20ths  ?  How  many  lOOths  ? 

Show  that  we  may  express  the  same  fractional  values  by  large 
numbers  or  by  small  numbers,  provided  that  we  do  not  change  the 
ratio  of  the  numerator  and  denominator. 

40.  Write  a  fraction  whose  denominator  is  5  times  its 
numerator  and  reduce  it  to  lowest  terms. 

41.  Write  several  fractions  whose  denominators  are  just 
twice  the  numerators.  To  what  fraction  is  each  one  of 
those  fractions  equal  ? 

42.  Change  J  to  some  equivalent  fractions.  Change  ^ 
to  some  equivalent  fractions.  Change  i  to  some  equiva- 
lent fractions. 


236  FRACTIONS 


43.  By  what  number  must  both  terms  of  the  fraction 
I  be  multiplied  to  change  it  to  -I  ?     Which  is  the  greater, 

44.  Change  |  to  15ths  and  tell  by  what  number  you 
multiplied  both  terms.  How  do  you  find  out  by  what 
number  both  terms  must  be  multiplied  ? 

45.  Change  the  following  fractions  to  higher  terms  and 
tell  in  each  case  by  what  number  you  multiplied  each  term  : 
f  to  21sts.    4  to  12ths.    I  to  20ths.    f  to  24ths.    |  to  28ths. 

46.  Sometimes  we  let  x  stand  for  a  number  that  we  are 
trying  to  find.  Write  out  the  following,  putting  the  true 
number  in  the  place  of  x: 

6  _Jr_  3. JL-  5. _x  A ^  7     «■  4  r 

y~~2  1*         4~l0-         8  ~"  2¥*         9  ~  l¥'         11~~¥^'         5  ~~  3T* 

X  is  no  more  difficult  in  this  place  than  the  interrogation  point. 

47.  Class  Exercise.  —  One  of  the  girls  may  give  a 
fraction.  One  of  the  boys  may  mention  a  higher  denomi- 
nator that  it  may  have,  and  the  class  may  change  it  so 
that  it  has  that  denominator. 

48.  Which  is  the  greater,  ^  of  a  foot  or  ^  of  it?  ^  of 
an  apple  or  y\  of  it  ?     -I-  of  a  dollar  or  -f^^  of  it  ? 

49.  y^o"  of  anything  is  what  %  of  it  ? 

50.  Which  is  greater,  |-  of  a  sum  of  money  or  49%  of  it  ? 

51.  Change  \  to  hundredths,  and  write  it  as  %.    Change 

f  to    %.        \  to    %.        4   to    %.        yV  to  %.        yV  to    %.        y9^  tO    %. 

52.  Class  Exercise. may  give  a  fraction  that 

he  can  change  to  lOOths  or  %,  and  some  one  else  may 
change  it. 

53.  Change  to  18ths  :  |,  \,  -J,  ^.  Why  can  you  not 
change  these  fractions  to  17ths  ? 

54.  Find  a  denominator  to  which  all  these  fractions  can 
be  changed,  and  change  them  :  \,  f,  f. 


FRACTIONS  237 

55.  The  least  common  multiple  of  the  denominators  is 
the  most  convenient  denominator.  What  is  the  least 
common  multi^^le  of  the  denominators  of  \,  |,  and  J? 
Chansre  these  fractions  to  12ths. 

56.  Find  the  least  common  multiple  of  tlie  denomi- 
nators of  the  fractions  ^  and  f ,  and  reduce  them  to  equiva- 
lent fractions  that  have  it  for  their  denominator. 

57.  Give  a  common  denominator  to  |,  |,  and  |  without 
changing  their  values. 

58.  Without    changing    values   give   common  denomi- 

nators  to  -3,  4?  5"'  ^^  "s?  T'  to^*  ^-^^  4'  ¥'  e^  12*  -'-^'  s?  5>  is* 
Tn  3    12    5        Tn  ii    1     1        Tn  1   ^   ^L       To  i   ^        To  K  +. 

_9_  q^o  -8     J^    1  ^Pn    5      5      2      3 

10*  -"-^-^    9'    27'    3*         -*-^    8?    6'    3'    4* 

59.  Class  Exercise. may  write  on  the  board 

two  fractions,  neither  of  whose  denominators  is  greater 
than  12,  and  the  class  may  change  them  to  the  same 
denominator  without  changing  the  value  of  either  of  them. 

60.  As  we  raise  fractions  to  higher  terms  by  multiply- 
ing both  terms  of  the  fraction  by  the  same  number,  so  we 
may  bring  them  to  lower  terms.     How  ? 

61.  Write  the  true  number  in  the  place  of  x  : 

4   X         18   X        J^O   —  £.        12   —  _J^        3_0   —    X         11  =i        28:=:iL 

"8~2"'       2T  T*       18~~9*       22~11*       40  4'       24  8*       3T  5* 

62.  Reduce  the  following  to  lower  terms,  and  tell  by 
what  number  you  divide  the  terms  of  each  fraction  : 

4         7        3       _3        __5_      JL      JL       6.      11      2.5       3.5.      _7JL      _2_5_      _95_ 
"J'     If     9'     11'     15'     10'     11'     8'     2  1'     30'     40'     100'     100'     100* 

63.  What  number  must  both  terms  of  ||-  be  divided  by 
to  reduce  that  fraction  to  lowest  terms  ? 

64.  What  is  the  largest  number  that  will  divide  both 
22  and  33.  What  name  do  we  giye  to  the  largest  number 
that  will  divide  two  numbers  ? 


238  FRACTIONS 

65.  Divide  both  terms  of  -f^  by  their  greatest  common 
divisor.  Which  is  the  greater,  the  fraction  you  get  or  ^q? 
Wliich  is  in  higher  terms  ? 

66.  Divide  botli  terms  of  ||  by  their  greatest  common 
divisor.      What  liave  you  done  to  the  fraction  ||? 

67.  Reduce  the  fraction  -^^  to  its  lowest  terms.  What 
number  is  the  greatest  common  divisor  of  both  terms  ? 

68.  Reduce  to  lowest  terms,  and  tell  what  common 
divisor  you  use  with  each  fraction  :    |,  |-|,  i|,  l^|,  |,  -^^^ 

_^4_    _8_     20     10     10      2.8 
16'   3  2'   32'    16'   32'    32 ' 

69.  If  you  cannot  see  the  greatest  common  divisor  at 
first,  and  if,  after  dividing,  your  fraction  is  not  in  its 
lowest  terms,  Avhat  "can  be  done  about  it  ? 

70.  Reduce  to  lowest  terms  : 

15.    11     2  5     4  8.     3  0     2.1    18     21    _SJ)_    _5_5        3_5 
30'   30'   30''   60'   45'   81'   36'    54'    100'    10  0"'    100' 

71.  Change  50%  to  a  fraction  in  its  lowest  terms. 

72.  Change  to  a  fraction  in  its  lowest  terms:  25%,  75%, 
20%,  40%,  00%,  80%,  30%,  70%,  90%,  45%,  r,n%,  35%. 

73.  Which  is  more,  15%  of  a  dollar,  or  ^^  of  it? 
23%  of  a  dollar,  or  i  of  it?     31%  of  a  dollar,  or\3_  of  it? 

74.  Class  Exercise. may  name  a  number  of 

%,  and  the  class  may  reduce  the  expression  to  a  fraction 
in  its  lowest  terms. 

75.  Can  you  reduce  f  to  lower  terms  ?     Explain. 

76.  How  do  you  reduce  a  fraction  to  lower  terms  ? 

77.  What  is  the  use  of  reducing  fractions  to  lower 
terms  ? 

78.  Write  -^  in  its  lowest  terms,  and  then  change  it  to 
14ths. 


FRACTKJNS  239 

79.  Put  ^  into  its  lowest  terms,  and  then  change  it  to 
21sts. 

80.  Bring  |  to  its  lowest  terms,  and  then  to  12ths. 

81.  Change  y\  to  its  lowest  terms,  and  then  to  lOOths. 

82.  Change  ^^  to  its  lowest  terms,  and  then  to  %. 

83.  Change  to  its  lowest  terms  and  then  to  %  :   |,  i|, 

4       21      i  J.      6  6 
T6>    3  6'     4  4'    T8' 

84.  What  is  the  ratio  of  15  to  20  expressed  in  its 
lowest  terms  ? 

85.  Express  in  its  lowest  terms  the  ratio  of  4  oz.  to  a 
lb.     8  oz.  to  a  lb.     12  oz.  to  a  lb.     14  oz.  to  a  lb. 

86.  Give  ratio  in  lowest  terms  of  18  to  20.  28  to  21. 
30  to  35.  40  to  50.  50  to  40.  45  to  50.  18  to  27. 
45  to  36.     72  to  84.     16  to  20.     30  to  24.     48  to  54. 

87.  The  flag  of  Company  E,  159th  Reg.  Ind.  Vol.,  is 

72  inches  long  and  54  inches  wide.     Express  the  ratio  of 

its  width  to  its  length  in  lowest  terms.     Express  the  ratio 

of  its  length  to  its  width. 

It  is  sometimes  well  to  let  the  children  take  sides  and  see  who  can 
stand  the  longest  without  failure  —  giving  fractions  and  reducing 
them  to  the  lowest  terms  or  to  higher  terms. 

88.  How  many  whole  ones  in  |  ?     Y-  ?    ¥  ^    "^  ^    ¥  ^ 

89.  A  fraction  whose  numerator  is  equal  to  or  greater 
than  its  denominator  is  called  an  Improper  Fraction. 
Find  some  improper  fractions  on  page  156. 

Show  pupils  that  an  improper  fraction  is  merely  a  form  of  division 
with  which  they  have  been  working  for  a  long  time. 

90.  Write  an  improper  fraction  whose  numerator  is  10, 
and  find  its  value. 

91.  Class  Exeiicise. may  name  an  improper 

fraction,  and  the  class  may  tell  its  value. 


240  FRACTIONS 

92.  What  kind  of  a  fraction  is  |-  ?    What  does  it  equal  ? 

93.  A  number  that  consists  of  a  whole  number  and  a 
fraction  is  called  a  Mixed  Number,  as  1^.  Give  some 
other  mixed  numbers. 

94.  In  a  mixed  number,  the  whole  number  is  called  the 
integral  part,  and  the  fraction  is  called  the  fractional  part. 
Give  the  integi'al  part  of  the  mixed  number  3|-.  Of  7^. 
Of  6i.     Of  144i. 

95.  Give  some  other  mixed  numbers  and  tell  which  is 
the  integral  and  which  is  the  fractional  part  of  each. 

96.  Give  a  mixed  number  the  integral  part  of  which  is  7. 

97.  Give  a  mixed  number  whose  fractional  part  is  f . 

98.  Can  you  see  any  reason  why  a  number  that  is 
made  of  a  whole  number  and  a  fraction  is  called  a 
''  mixed  "  number? 

99.  Change  the  improper  fraction  ^  to  an  equivalent 
mixed  number,  or  find  how  many  whole  ones  in  |. 

100.  Change  to  an  equivalent  mixed  number  :  J,  li,  i^, 

9     _1_1      1.5     _2_4      3_0     _2_2     _1_7_     2. 5.     1  6     _5_5.    J_3     4X    _5_9      4  9 
7 J      5"?    1  U     7   '    1  1'      3"?     5  ?    12^  ~Y~J     9  '     8  ?     5  ?     8  J    12* 

101.  Look  at  the  definition  of  an  improper  fraction  in 
Ex.  89  and  tell  whether  or  not  f  is  an  improper  fraction. 
How  many  whole  ones  does  it  equal  ? 

102.  Can  you  change  the  improper  fraction  J^^-  to  an 
equivalent  mixed  number?     Explain. 

103.  Change  to  equivalent  whole  numbers  the  following 
imnrnnpr  fraction'^  •    18    2  4     21     4  8     4  8     4  8    4 8.    7_2    _6 4_    3_6_   _6_3. 

104.  Write  some  fractions  whose  denominators  are  each 
7  and  whose  numerators  are  multiples  of  7,  and  change 
them  to  equivalent  whole  numbers. 

105.  Ag^  of  a  pie  are  equal  to  how  many  whole  pies? 

18?         3Q.?        4_2  9         72  ? 
6' •  F^  •         '6     •  6     • 


FRACTIONS  241 

106.  Write  some  fractions  whose  denominators  are  each 
8  and  whose  numerators  are  multiples  of  8,  and  reduce 
them  to  equivalent  whole  numbers. 

107.  Class  Exercise. may  give  an  improper 

fraction  that  can  be  reduced  to  a  whole  number,  and  the 
class  may  reduce  it. 

108.  Give  an  improper  fraction  that  can  be  reduced  to 
a  mixed  number  and  reduce  it. 

109.  Reduce  to  equivalent  mixed  numbers  :  ^,  ^J-,  -L®-,  ^^, 

6  4     9  9      4i)_     S2_     5.1     4  7     _8 1 
~9  ">    1  2  J  "  7  J     9  J    1  2  J     o  ?    1  1  • 

110.  Class  Exercise. may  give  an  improper 

fraction  that  can  be  reduced  to  a  mixed  number,  and  the 
class  may  reduce  it. 

111.  Reduce  to  whole  or  mixed  numbers  :  -y-,  ^gS  ^^^ 

6  1      4_8      4  8      4_8_      5_0      JJ)      3JL      6  1 
12'      6  '     12'      8  '      9  '      8~'    "5  '     T"' 

112.  Tell  how  you  reduce  an  improper  fraction  to  a 
whole  or  mixed  number. 


113.  Reduce  the  following  to  equivalent  mixed  numbers 

h\r   Inner  rliviQinn  •  -2-99   683   849   476   1000   8246   833   789 

114.  How  many  7ths  in  5  whole  units  ?  In  9  whole 
units  ?  In  11  ?  In  13  ?  What  kind  of  fractions  have 
you  been  changing  these  whole  numbers  into  ? 

115.  How  many  8ths  of  an  inch  in  3  inches?  6  in.? 
9  in.?  4  in.?  12  in.?  Into  what  form  have  you  been 
chano^ing-  these  whole  numbers  ? 

116.  Change  5  into  an  improper  fraction  whose  denomi- 
nator is  10,  or  find  how  many  lOths  in  5. 

117.  Change  6  into  an  improper  fraction  whose  denomi- 
nator is  8.  Into  one  whose  denominator  is  5.  Into  one 
whose  denominator  is  7. 

HORN.    ARITH. 16 


242  FRACTIONS 

118.  Change  4  to  6ths.  To  8ths.  To  lOths.  To  9ths. 
To  12ths. 

119.  Class  Exercise.  —  John  may  name  a  whole  num- 
ber, and  the  class  may  reduce  it  to  an  improper  fraction, 
with  a  denominator  that  Mary  may  choose. 

120.  How  many  7ths  in  2 1  ?  In  4f  ?  In  5f  ?  In  8f  ? 
In  6^  ?  Into  what  form  have  you  been  changing  these 
mixed  numbers  ? 

121.  Change  into  equivalent   improper   fractions  :    5|-, 

122.  Write  the  following,  putting  the  true  numbers  in 
the  place  of  x:    7i  =  f.      3f=f.      6i  =  |-.     5f  =  f.     ^=^. 

i    2     _J_  Q3   £.  C7   X  73     X 

^11   —   11'        fJrj  —   rj.        o^  —  -g-.         'TT  —  TT* 

123.  Give  a  mixed  number  whose  fractional  part  is  ^, 
and  reduce  the  mixed  number  to  halves. 


124.  Give  a  mixed  number  whose  fractional  part  is  J, 
and  reduce  the  mixed  number  to  an  equivalent  improper 
fraction. 

125.  Give  a  mixed  number  Avhose  integral  part  is  4,  and 
reduce  it  to  an  equivalent  improjjer  fraction. 

126.  How  do  you  reduce  a  mixed  number  to  an  im- 
proper fraction  ? 

127.  Reduce  to  equivalent  improper  fractions  tlie  fol- 
lowing :  12i,  621  173^  411^  lej,  33^,  871  37^,  66|,  31J. 

128.  Class  Exercise. may  put  a  list  of  mixed 

numbers  on  the  board,  and  the  class  may  reduce  them  to 
improper  fractions. 

129.  yV  of  a  foot  -f-  3^  of  a  foot  -|-  |J  of  a  foot  -f  y^^  of  a 
foot  equals  how  many  feet  ? 

130.  -J  of  anything  +  |  of  it  +  |  of  it  =  what  part  of  it? 


FRACTIONS  243 

131.  Write  four  fractions  whose  (leuoininator  is  7  imd 
tincl  their  sum. 

132.  Class  Exercise. may  give  three  fractions 

having  the  same  denominator,  and  the  class  may  find 
their  sum.  If  the  sum  is  an  improper  fraction,  reduce 
it  to  a  whole  or  mixed  number. 

133.  From  ^  take  tV     ^  -  t%  =  ?     A-t\=? 

134.  Recopy  Fig.  2,  p.  233,  and  find  from  it  how  many 
32ds  of  it  J  +  3V  equal. 

135.  Find  from  Fig.  2  the  values  of  x  in  the  following 
equations,    and   write    them    in    place    of   it:    \+^2—T2' 

3_i__l    — JL.         1 -U—l— —    ^_         •5_1_-J_ — _=^_        5._|_JL — _^_        I_|__l_  —  ^■^_ 
4    "    32~~32'        8    '32        ^2*        8    '32         32"       8^32        32*       8^32"~32- 

1     _i_     1_  —  _x_  _3 I 1_  —  _Jc_  _5 I 1_  —  _i_  _7__  J !__  —  _-^_ 

T6"'T""3^2~32*  1G^32~32-  1G^3  2~32*  16'32~32- 

9       I       1     _^_  iO  J 1_  —  _x_  iJi  J 1_  —   jc_  JL2.     I 1_  —  _x_ 

16~'~32  —  32"  10"  ^32  32"  16'32  3  2*  16132  3  2' 

136.  How  much  does  }f  -f  3^2^  of  Fig.  2  lack  of  being 
the  whole  figure  ? 

137.  Change  each  of  the  plus  signs  in  Ex.  135  to  a 
minus  sign,  and  write  the  equations  again,  putting  in  the 
true  numbers  in  place  of  x. 

138.  Can  you  find  a  shorter  way  of  adding  or  subtract- 
ing fractions  than  by  dividing  a  figure  and  counting  the 
parts  ? 

Bring  out  the  idea  that  fractions  must  be  reduced  to  the  same 
denominator  before  adding  them  or  subtracting  one  from  another. 

139.  To  change  to  32ds  by  what  number  must  both 
terms  of  ^  be  multiplied  ?    ^  ?    ■^? 

140.  Find  values  of  x  in  the  following  equations,  first 
reducing  all  fractions  to  32ds : 

l_Lll— ^L.        JL-I-U- —   ^  3._1_L  — _*_       i_Lii  — _*_       3.4_JL1.  — _*_ 

'^T'a^  —  S2*         4IR2  —  ■rT*         4         S2  —  S2'         8"^  3  2  —  S'2*        Ri^ 


■S"   '    32  —  32*        4    '^32  —  ST*        4        32  —  32'        8    "^  3  2  —  32*       8    "^  3  2        32* 

5  JJ, _x_         7.  I_L  —  _=?_         J 1-11  —  _=L-         _3 L  11  —  J?_ 

8  32  —   32"        8  32  —  32*        16'32  32"        16'32  32* 


244  FRACTIONS 

141.  How  many  inches  in  1  foot  and  3  inches  ?  What 
must  you  do  with  the  1  foot  before  you  can  add  3  inches  ? 

142.  To  add  ^  and  ^,  what  must  you  change  ^  into  ? 
Why  not  change  ^  to  halves  ?     2  +  i~  -^     i  ~  i~- 

143.  To  add  or  subtract  4ths  and  8ths,  what  common 
denominator  must  they  have  ?     4  +  8~^     i~l~^ 

144.  Can  you  see  the  use  of  learning  to  reduce  frac- 
tions to  higher  terms  ? 

145.  Find    values    of   x   in    the    following:    1  +  1  =  |. 

2  ^  6  ~  6*        3^6~6'        3^6""6-        2         3        6*         2         6        6* 

146.  Draw  a  circle,  divide  it  into  6ths,  and  show  that 
your  work  was  right  in  the  preceding  example. 

147.  Divide  the  circle  into  12ths,  and  prove  your  work 
after  you  have  found  the  values  of  x  in  the  following  : 

1  _i    _L  _iL-         1  J 1_  _^         X  J 1_  _i^         1  J L_  ^— 

2  "^  12  ~   12'         4  "1"   12  "~   12*        3  "1"  12  "~   12'        6     '     1 2  ~"  12* 

3  "1"  4  ~  T2-        4.  ~   12  ~~  T2-        3  ~   12  ~~  ^2'        6   ~   1 2  ~"  12* 

6     I      _1_  _  ^l_  2     I     _1_  _  _^«;L  5 1_  —  _±^  2. 1^  —  _^_ 

6  ^   12  ~~   12'        3     '     12  ~"   12*        6  12  ~~   12'        3  1 2   ~~  12* 

3  ^  f         6  ■"   12-  3         4  ^  12  ~~   12-  3         4^6  12' 

4  6  ^  12  ~   12*  6  4         3  ~~   12*  6  3  ^  12  12* 

148.  Draw  a  rectangle  5  inches  long  and  2  inches  wide, 
and  prove  your  work  after  finding  the  values  of  x  in  the 
following  equations  : 

111    X  1    1     X  1 1      _x  _3 1.  —  _x_ 

o"  "I     2"  ~  T"5'*  "5         T^  ~  Tiff'  2"         10~~10"  10         5~10* 

149.  Divide  each  square  of  your  rectangle  into  halves, 
and  prove  your  results  in  the  following  : 

1    i_    1    —    X  1 L  —  _*_  _1 I 1-  —  -J?_  1 3L  —  _*- 

2'^20  —  20'  2  20  —  20'  10'20  20*  Z         20  20* 

150.  In  adding  |  and  -j^  why  do  you  reduce  ^  to  20ths  ?, 

Lead  the  children  to  observe  that  in  all  this  concrete  work  they 
have  used  as  a  common  denominator  the  number  that  is  th^  le^st 
con^uiou  nmltiple  of  the  denominators, 


FRACTIONS  245 

151.  What  is  the  least  number  that  will  contain  8  and  3  ? 
Change  ^  and  \  to  24ths,  and  find  their  sum.  Find  their 
difference. 

152.  Change  ^  and  f  to  a  common  denominator,  and  add 
them.     Find  their  difterence. 

153.  Change  to  a  common  denominator  and  add  : 

2nndl         1-1-1         1-1-2  3ii         i-1-2         _3_i2         i4_3 

154.  Philip  lost  y  of  his  money  and  spent  i  of  it.  What 
part  of  it  had  he  left?  If  he  had  il4  to  begin  with, 
how  much  had  he  left  ? 

155.  Fred  took  a  bicycle  trip  from  his  home  to  Indian- 
apolis. In  the  first  5  days  he  rode  j\  of  the  distance.  On 
the  6th  day  he  rode  -^  of  the  distance.  What  part  of  the 
distance  had  he  still  to  ride  ? 

156.  Arthur  spent  ^  of  his  money  at  one  time,  and  gave 
^  of  it  at  another  time.  What  part  of  it  did  he  spend  ? 
What  part  of  it  had  he  left  ?  If  f  2  was  what  he  had  left, 
how  much  had  he  at  first  ? 

157.  Mrs.  Sampson  spends  |  of  the  money  she  receives 
as  interest  for  board  and  ^  of  it  for  clothes.  What  part 
of  it  has  she  left  ? 

158.  Mr.  Perkins  laid  off  J  of  an  acre  for  turnips,  ^  of  an 
acre  for  tomatoes,  and  ^  of  an  acre  for  peas.  How  many 
acres  did  he  lay  off  for  all  ? 

159.  Write  two  fractions  that  can  be  reduced  to  20ths, 
and  find  their  sum. 

160.  Write  two  fractions  that  can  be  reduced  to  SOths, 
and  find  their  difference. 

161.  A  lady  spent  ^  of  her  money  on  Monday,  and  i  of 
it  on  Tuesday.  What  part  of  her  money  did  she  spend, 
and  what  part  had  she  left  ?  If  at  first  she  had  i  18,  how 
much  did  she  spend  on  Monday  ?     On  Tuesday  ? 


246  FKACTIONS 

162.  George  spent  ^  of  his  money  for  a  watch,  and  ^  of 
it  for  a  coat.  What  part  did  he  spend  and  what  part  had 
he  left  ?  If  he  had  $  20  to  begin  with,  how  much  had  he 
left? 

163.  Out  of  a  flock  of  chickens  ^  died,  \  were  sold, 
and  Yo  were  lost.  What  part  of  them  were  left  ?  If 
there  were  20  chickens  in  the  tirst  place,  how  many 
remained  ? 

164.  Find  difference  of  ^  and  |.     Find  their  sum. 

165.  A  man  bought  -J  of  an  acre  of  land,  and  sold  ^  of 
an  acre  to  his  brother.    What  part  of  an  acre  did  he  keep? 

166.  Mrs.  Miller  paid  I  of  a  dollar  for  some  butter,  i  a 
dollar  for  some  coffee,  f  of  a  dollar  for  some  sugar,  and 
had  ^  a  dollar  left.     How  much  had  she  at  first  ? 

167.  A  milkman  left  ^  of  a  gallon  of  milk  at  one  house, 
f  of  a  gallon  at  another,  |  of  a  gallon  at  another.  How 
many  gallons  of  milk  did  he  have  in  all  ? 

168.  A  field  is  J  of  a  mile  long  and  i  of  a  mile  wide. 
What  fraction  of  a  mile  is  the  difference  between  its 
length  and  its  width  ? 

169.  Irene  spent  J  of  an  hour  in  school  in  writing,  |^  of 
an  hour  in  preparing  her  geography  lesson,  and  J  of  an 
hour  in  reciting  it.      How  much  time  did  she  spend  in  all  ? 

170.  If  the  session  of  school  was  3  hours  long,  how 
much  time  had  she  left? 

171.  Find  sums  : 


^  8|   ^n    8f   If 

2f      .3f        2,V      7|      4f 

C)5 

4f 

8f 

8t\ 
4^ 

172.    Find  differences  : 

8|         5i          9|          8| 
4i          Sf         2|         3f 

6| 
If 

5i 
If 

6f 

8t\ 
4-7- 

FRACTIONS  247 

173.  Mr.  Turner  had  83-|-  acres  of  wheat,  78^  acres  of 
corn,  and  13|-  acres  of  oats.  How  many  acres  had  he  in 
cultivation  ? 

174.  Mr.  Green's  Jersey  cow  Bova  gave  milk  enough  to 
make  17|^  lb.  of  butter  in  one  week,  18|  lb.  the  next  week, 
19^  lb.  the  next  week,  and  18J  lb.  the  next  week.  How 
much  was  her  average  weekly  yield  of  butter  ? 

175.  Bova's  price  was  $575.  She  and  her  calf,  Good 
Boy,  were  sold  for  $600.  What  Avas  the  price  of  the 
calf  ?     Find  its  ratio  to  the  price  of  the  cow. 

176.  Make  problems  in  which  fractions  are  added. 

177.  Find  differences : 

From     8iorT\      94      H     ^     9f     ^     ^     8f      7J 
take      3i  or  A     ?I?i?iilli?l!lli 

178.  H  the  bread  that  you  eat  in  1  day  requires  4  oz.  of 
flour  to  make  it,  how  many  oz.  of  flour  will  you  eat  in  a 
year  of  365  days  ?     How  many  lb.? 

179.  196  lb.  of  flour  make  a  barrel.  If  3^ou  ate  5  oz.  of 
flour  each  day,  how  much  less  than  a  barrel  would  you  eat 
in  a  leap  year  ? 

180.  John  rode  on  his  bicycle  to  a  town  28  miles  away. 
He  stopped  to  rest  and  found  he  had  traveled  9|  miles. 
How  much  farther  had  he  to  go  ? 

181.  After  riding  6|  miles  farther,  how  many  miles  re- 
mained ? 

182.  Multiply  6  by  \.  To  multiply  6  by  \  we  take  \  of 
6.     Multiply  24  by  \.     18  by  i.     27  by  J.     30  by  yV- 

183.  Multiply  I  by  |-.  To  multipl}^  i  by  |-  we  take  ^  of 
\.     How  much  is  |-  of  ^  ? 

1QJ.        Iv5— ?        4v'5— ?        _6_v2J.— ?        7.vi6._?        15  y  11—9 
184.      8'^T—-       "S-^y— •       TT^~9^— •       ¥^2  1  —  -       T^'^TZ      ■ 

Iv2v3_9         .5v7v6_'>         2v6v2-l—  9        5.v6v^5.  —  9 


248  FRACTIONS 

185.  Class  Exercise. may  give  some  fractions, 

and  the  class  may  find  their  product. 

186.  Give  an  improper  fraction  and  tell  what  an  im- 
proper fraction  is. 

187.  A  fraction  that  is  not  improper  is,  of  course,  proper. 
-|  is  a  proper  fraction.  Compare  its  numerator  and  de- 
nominator and  tell  why  it  is  a  proper  fraction. 

188.  Give  a  proper  fraction  whose  denominator  is  7. 
Give  three  other  proper  fractions  that  express  7ths. 

189.  Give  four  proper  fractions  that  express  llths. 

190.  Multiply  a  proper  fraction  by  an  improper  fraction. 

191.  Find  the  product  of  two  improper  fractions  and 
reduce  this  product  to  a  whole  or  a  mixed  number. 

192.  Multiply  a  proper  fraction  whose  denominator  is 
an  odd  number  by  a  proper  fraction  whose  denominator  is 
an  even  number. 

193.  Is  99%  a  proper  or  an  improper  fraction? 

194.  Reduce  the  mixed  number  2^  to  9ths  and  tell  how 
you  reduced  it. 

195.  I  wish  to  multiply  the  mixed  number  2^  by  J. 
What  must  be  done  to  the  mixed  number  so  that  it  may 
be  multiplied  by  a  fraction  ? 

196.  Reduce  the  mixed  number  8J  to  an  improper  frac- 
tion and  multiply  it  by  ^. 

197.  Reduce  the  mixed  numbers  2^  and  2^  to  improper 
fractions  and  find  their  product. 

198.  Reduce  the  following  mixed  numbers  to  improper 
fractions  and  find  values  of  x : 

2-1  x^l    =x.      21  X  2|  =  X.     3f  X  2^=x.     4|  x  4^  =  x. 
•y-  X  l-j\  =  x.      3i  X  24  =  x.     8i  X  H=x.     5f  X  3j  =  x. 


FRACTIONS  249 

199.  How  much  will  2|  pounds  of  soap  cost  at  12|^  cents 
a  pound  ? 

Find  the  cost  of: 

200.  5^  tons  of  hay  @  12|-  dollars  a  ton. 

201.  41  quarts  of  strawberries  @  8^  cents  a  quart. 

202.  3|  acres  of  land  @  62|-  dollars  per  acre. 

203.  IJ  dozen  pencils  @  33^  cents  per  dozen. 

204.  3|-  quarts  of  milk  @  6^  cents  per  quart. 

205.  5|  yards  of  carpet  @  66|^  cents  per  yard. 

206.  A  whole  number  is  called  an  Integer,  as  4,  10,  etc. 
Name  two  integers  and  give  their  product.  Is  the  prod- 
uct an  integer  or  a  fraction? 

207.  Multiply  2V  ^y  ~V"-  Does  it  make  any  difference 
in  the  result  whether  that  multiplier  is  called  16  or  ^-? 

208.  Multiply  I  by  10.  j\  by  22.  -f^  by  7.  |  by  6. 
^Vbyie.      A  by  21.      ^^hj20.      ^3  by  27.      if  by  12. 

209.  Multiply  18  by  -^.     24  by  ^5.    42  by  f     81  by  -|. 

210.  Write  a  fraction  and  multiply  it  by  an  integer. 

211.  Multiply  an  integer  by  a  fraction. 

212.  When  you  wish  to  multiply  a  mixed  number  by 
an  integer  or  an  integer  by  a  mixed  number,  do  not 
reduce  either  of  them  to  an  improper  fraction. 

Multiply  li  Multiply  12 

by  8_  by  _3i 

Why  is  it  best  not  to  reduce  either  of  the  numbers  to 
an  improper  fraction  ? 

213.  How  many  rods  in  2|-  miles  ?    3^  miles  ?    7|-  miles  ? 

214.  How  many  minutes  in  3|-  hours  ?  7f  liours  ? 
Ij  hours  ? 


250  FRACTIONS 

215.  At  12|-  cents  a  yard,  how  much  will  8  yards  of 
lace  cost  ? 

216.  At  37|-  cents  a  yard,  how  much  will  15  yards  of 
sheeting  cost  ? 

217.  Tell  how  you  multiply  a  mixed  number  by  an 
integer. 

218.  ^Multiply  f  by  itself. 

OIQ        ^nncjTP  •      3.      4       5.     _3_     _7_      3      J_0.     A      1     ii 

220.  bquare :    I3,  l>2^,  2^,  Ig^,  l^^,  ly,  z^,  42^,  o^. 

221.  How  many  yards  long  is  a  rod  ?     Tell  how  you 

find  the  number  of  square  yards  in  a  square  rod. 

Let  the  square  rod  with  its  divisions  be  drawn  on  the  floor  of  the 
schoolroom  and  remain  until  it  is  worn  off. 

222.  How  many  square  yd.  in  a  sq.  rd.  ?  In  f  of  a 
sq.  rd.  ?  In  y\  of  a  sq.  rd.  ?  In  3%  of  a  sq.  rd.  ?  In  || 
of  a  sq.  rd.  ? 

223.  How  many  ft.  long  is  a  rod  ?  How  many  square 
feet  in  a  square  rod  ? 

224.  How  many  sq.  ft.  in  |  of  a  sq.  rd.  ?  In  -^j  of  it  ? 
In  if  of  it  ?     In  tIt  of  it  ? 

225.  How  many  yd.  in  the  perimeter  of  a  sq.  rd.  ? 
How  many  ft.  ? 

226.  How  many  feet  of  fence  will  it  take  to  inclose 
a  burial  lot  2  rd.  square  ?  How  much  will  it  cost  at 
50  cents  a  foot  ? 

227.  How  many  yd.  of  fence  will  inclose  a  lot  8  rd. 
square  ?     How  much  will  it  cost  at  11.75  a  yd.  ? 

228.  Write  an  improper  fraction  whose  denominator  is 
5,  and  change  it  to  a  whole  or  a  mixed  number. 

229.  Draw  a  line  J  of  an  inch  long,  and  see  how  many 
times  a  line  i  of  an  inch  long  is  contained  in  it.    ^  ^  4  =  ^ 


FKACTIONS  251 

230.  How  many  times  is  |  of  a  pie  contained  in  -i-  of  a 

|jiC  .         2*6  ' 

231.  Turn  to  Fig.  1  and  show  \  of  it.  Show  Jg-  of  Fig.  1. 
How  many  times  is  Jg  contained  in  J  ?     ^  ^  Jg-  =  ? 

232.  Each  triangle  is  what  part  of  Fig.  3? 
Show  \  of  Fig.  3.  Show  |  of  it.  How  many 
times  is  |-  contained  in  i  ?     i  -^  i  =  ? 

Show  that  in  dividing  one  fraction  by  another,  the 
same  result  is  obtained  by  inverting  the  divisor  and 
multiplying,  as  by  actually  measuring  off  one  part  of 
an  object  upon  another  part,  and  counting  the  measurements. 

233.  Divide:  fbyf.   |  by  f .  f  by  4.   1^^=^^  A-^-f  =  ? 

234.  Write  a  fraction  whose  denominator  is  7,  and 
divide  it  by  a  fraction  whose  denominator  is  14. 

Class  drills  like  the  following  are  useful :  "  Take  |,  multiply  it  by 
4,  add  I,  reduce,  add  ^,  change  to  improper  fraction,  divide  by  8, 
square,  nmltiply  by  5,  add  ^,  reduce,  divide  by  3,  subtract  y  etc. 

235.  Write  an  improper  fraction  and  divide  it  by 
another  improper  fraction. 

236.  Write  an  improper  fraction  and  divide  it  by  a 
proper  fraction. 

237.  Write  a  proper  fraction  and  divide  it  by  another 
proper  fraction. 

238.  Write  a  mixed  number,  reduce  it  to  an  improper 
fraction,  and  divide  it  by  some  other  fraction. 

239.  Reduce  3|^  to  an  improper  fraction,  and  divide  it 

byA- 

240.  Divide:    2|byf.     2f  by  j^.     4fby|.     9|by^. 

241.  Reduce  to  improper  fractions  and  divide  :  9J  by 
3i      6iby2f     16|by6J.     7|  by  2|.     8|  by  8J. 


252  FRACTIONS 

242.  Reduce  mixed  numbers  to  improper  fractions,  and 
find  values  of  x  : 

^^^^x,       7y\-^5f  =  2;.      8|^13J=:r.       333^-21=2:. 

243.  Tell  how  you  divide  one  fraction  by  another. 

244.  At  2|-  cents  apiece,  how  many  oranges  can  be 
bought  for  15  cents  ?     25  cents  ?     40  cents  ?     50  cents  ? 

245.  At  3|^  cents,  how  many  balls  can  be  bought  for  13i 
cents  ?     33|^  cents  ?     36|  cents  ?     43J  cents  ?     16J  cents  ? 

246.  At  6^  cents  per  yard,  how  many  yards  of  ribbon 
can  be  bought  for  25  cents  ?     50  cents  ?     75  cents  ? 

247.  At  8i  cents  per  pound,  how  many  pounds  of  rice 
can  be  bought  for  25  cents  ?     75  cents  ?     50  cents  ? 

248.  How  much  does  the  quotient  of  J  ^  ^-  lack  of 
being  equal  to  1  ? 

249.  1  is  how  much  greater  than  the  quotient  of  ^  ^ 

15  ? 

Ti  • 

250.  What  is  the  sum  of  |  and  f  ?  What  is  their  differ- 
ence ?  Product  ?  Quotient  of  greater  divided  by  less  ? 
Quotient  of  less  divided  by  greater  ? 

251.  Abraham  Lincoln  w^as  born  in  MDCCCIX.  How 
old  was  he  in  MDCCCLXV,  the  year  in  which  he  died? 


FRACTIONS  253 

DRY   MEASURE 

2  pints  (pt.)=  1  quart  (qt.). 
8  quarts  =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.). 

LIQUID   MEASURE 
4  gills  (gi.j)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts  =  1  gallon  (gal.). 

MEASURE    OF  TIME 

60  seconds  (sec.)=  1  minute  (min.). 
60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

12  months  =  1  year  (yr.). 

365  or  366  days    =  1  year. 

LINEAR   MEASURE 

12  inches  (in.)=  1  foot  (ft.)  . 
3  feet  =  1  yard  (yd.). 

5 J  yards  =  1  rod  (rd.). 

16|^  feet  =  1  rod. 

320  rods  =  1  mile  (mi.). 

AVOIRDUPOIS   WEIGHT 

16  ounces  (oz.)=  1  pound  (lb.). 
2000  pounds       =1  ton  (T.). 

SQUARE   MEASURE 

144  square  inches  (sq.  in.)=  1  square  foot  (sq.  ft.). 
9  square  feet  =  1  square  yard  (sq.  yd.). 

CUBIC   MEASURE 
1728  cubic  inches  (cu.  in.)=:  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  ^  1  cubic  yard  (cu.  yd.). 


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